Singularity formation and regularization at multiple times in the viscous Hamilton

Abstract

The Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation (VHJ) from stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after appearance of singularity, which occurs through gradient blow-up (GBU) on the boundary. Whereas viscosity solution theory has been extensively applied to many PDEs, there seem to be less results on refined singular behavior of solutions. Although occurrence of two types of GBU, with or without loss of boundary condition (LBC), are known, detailed behavior after GBU has remained open except for a strongly restricted special class of one dimensional solutions.

In this paper, in general dimensions, we construct solutions which undergo GBU with LBC arbitrarily many times and then recover regularity, as well as solutions without LBC at first GBU time. In one space dimension, we obtain the complete classification of viscosity solutions at each time, which extends to radial case in higher dimensions. Furthermore we show the existence of solutions exhibiting an arbitrarily given combination of GBU types with/without LBC at multiple times, in which a new type of behavior called “bouncing” is discovered.

Global weak solutions of VHJ with multiple times singularity turn out to display larger variety of behaviors than in Fujita equation. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires delicate arguments. Since we do not rely on any known special solution unlike in Fujita equation, our method is expected to apply to other equations. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. Our results imply that for certain spatial distributions of rewards, if a controlled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.

Keywords

Viscous Hamilton–Jacobi equation gradient blow-up loss and recovery of boundary conditions multiple time singularities viscosity solutions zero-number critical parameters stochastic control

1. Introduction and main results

1.1. Problem

We consider the initial boundary value problem for the viscous Hamilton-Jacobi equation: $\begin{array}{l} (1.1) & \{\begin{matrix} u_{t} - Δ u = | \nabla u |^{p}, & x \in Ω, t > 0, \\ u = 0, & x \in \partial Ω, t > 0, \\ u (x, 0) = ϕ (x), & x \in Ω \end{matrix} \end{array}$ with $p > 2$ , where $ϕ \in X : = {ϕ \in C^{1} (\overline{Ω}); ϕ ⩾ 0, ϕ = 0 on \partial Ω}$ . Throughout this paper, Ω is a smoothly bounded domain of $R^{n}$ .

By standard theory [19], problem (1.1) admits a unique, maximal classical solution u satisfying $u ⩾ 0$ in $\overline{Ω} \times [0, T)$ , where $T = T (ϕ) \in (0, \infty]$ denotes the maximal existence time. It is known that, for suitably large initial data, solutions blow up in finite time, i.e., $T (ϕ) < \infty$ (see, e.g., [1,2,38]). Since the solution satisfies $\begin{array}{l} {‖ u (t) ‖}_{\infty} ⩽ ‖ ϕ ‖_{\infty}, 0 < t < T, \end{array}$ by the maximum principle, the classical solution can only cease to exist through gradient blowup (GBU) $\begin{array}{l} lim_{t \to T} {‖ \nabla u (t) ‖}_{\infty} = \infty . \end{array}$ However the solution survives after the blow-up time and can be continued as a generalized viscosity solution of (1.1). More precisely, by [8], problem (1.1) admits a unique, global (generalized) viscosity solution; this solution is nonnegative and continuous in $\overline{Ω} \times [0, \infty)$ , and it coincides with the (unique) classical solution in $(0, T)$ . Throughout this paper, we shall also denote it by u, without risk of confusion (or $u (ϕ; \cdot, \cdot)$ when we need to emphasize its dependence on the initial data).

Whereas the theory of viscosity solutions (cf. [12]) has been extensively studied and applied to many partial differential equations, there seem to be less results on refined behavior of the weak solution. Our purpose is to investigate the behavior of the global viscosity solution u for $t > T (ϕ)$ . First of all, it is known that the solution u enjoys interior regularity, i.e. $u \in C^{1, 2} (Ω \times (0, \infty))$ , and solves the PDE in the pointwise sense in $Ω \times (0, \infty)$ , but it may lose the boundary conditions in the classical sense. Indeed such a possibility, which was suggested in [8], was confirmed in [33,36] where it was shown that, for suitably large initial data, the solution undergoes a loss of boundary conditions (LBC) at some times $t > T (ϕ)$ , i.e., $\begin{array}{l} sup_{x \in \partial Ω} u (x, t) > 0 . \end{array}$ However, some exceptional GBU solutions without LBC were also shown to exist in [33,34], found as separatrices between global solutions and GBU solutions with LBC. We see that solutions with LBC, which are meant to satisfy zero boundary conditions in the generalized viscosity sense (see [8] for precise formulation), nevertheless have to continuously take on some positive boundary values. This apparently paradoxical situation can however be interpreted in a more intuitive way, when one recalls that the global viscosity solution can also be obtained as the limit of a sequence of global classical solutions of regularized versions of problem (1.1), with truncated nonlinearity (see e.g. [34] and Section 2 below for details). Since this convergence is monotone increasing but not uniform up to the boundary, the loss of boundary conditions can in this framework be seen as a more familiar boundary layer phenomenon.

On the other hand, it was shown in [35] that u becomes a classical solution again for all sufficiently large time, i.e. there exists $\tilde{T} ⩾ T$ such that $\begin{array}{l} u \in C^{2, 1} (\overline{Ω} \times (\tilde{T}, \infty)), with u = 0 on \partial Ω \times [\tilde{T}, \infty), \end{array}$ and furthermore u decays exponentially in $C^{1} (\overline{Ω})$ as $t \to \infty$ . The natural and important question is thus to describe the behavior of u in the intermediate time range $[T, \tilde{T}]$ . In [34], for $n = 1$ and $Ω = (0, 1)$ , a strongly restricted special class of symmetric initial data was studied, for which the solution immediately loses the boundary conditions after T and is immediately and permanently regularized after recovering the boundary condition. It has remained open whether or not there exists a viscosity solution undergoing GBU at multiple times. We affirmatively answer the question in general dimensions constructing a viscosity solution exhibiting GBU with LBC at multiple times. We give a precise behavior in one dimension and in radial domains in higher dimensions. There have been no results on the fundamental behaviors like finiteness of GBU times and existence of waiting time of recovery of regularity, except the special case in [34]. We give a complete classification at each time. Furthermore for each $m ⩾ 2$ and arbitrarily given combination of GBU types with/without LBC at m times, we construct a viscosity solution undergoing this exact combination of GBU in the case of $n = 1$ , which can be easily extended to radial case with $n ⩾ 2$ .

In view of the very large literature devoted to the viscous Hamilton–Jacobi equation, this introduction makes no attempt to be exhaustive. For other aspects of (1.1) and related problems, we refer to, e.g., [2,4,5,9–11,13–15,17,21,23,24,32,37–39,41] and the references therein.

1.2. Applications to stochastic control problems

Let us recall that (1.1) arises in stochastic control problems. Namely, consider the controlled n-dimensional stochastic differential equation $\begin{array}{l} d X_{s} = α_{s} d s + \sqrt{2} d W_{s}, s > 0, with X_{0} = x \in Ω, \end{array}$ where the stochastic process ${(X_{s})}_{s > 0}$ represents the position or state of the system, ${(W_{s})}_{s > 0}$ is a standard Brownian motion and ${(α_{s})}_{s > 0}$ is the control (in other words, the controller can choose the velocity of X). The spatial distribution of rewards is given by a function $u_{0} \in C_{0} (\overline{Ω})$ . More precisely, at a given time horizon $s = t > 0$ , the final reward is $u_{0} (X_{t})$ if X stays in Ω until time t, and 0 otherwise. Finally, the cost of the control at each time s is assumed to be $k_{p} | α_{s} |^{q}$ as long as $X_{s}$ stays in Ω, where $q = p / (p - 1)$ is the conjugate exponent of p and $k_{p} > 0$ is a normalization constant. The goal of the controller is then to maximize the net gain $\begin{array}{l} G_{t} = χ_{τ > t} u_{0} (X_{t}) - k_{p} \int_{0}^{τ} | α_{s} |^{q} d s, \end{array}$ where τ denotes the first exit time of X from Ω. It is known (see [7,8,18] for details) that the maximal gain, also called value function of the stochastic control problem, is given by the unique global (continuous) viscosity solution u of (1.1), namely: $\begin{array}{l} u (x, t) = sup_{{(α_{s})}_{s}} E (G_{t} | X_{0} = x), \end{array}$ where $E (\cdot | X_{0} = x)$ denotes the conditional expectation with respect to the event ${X_{0} = x}$ , and the supremum is taken over all (admissible) controls. In this framework, the existence of solutions with multiple times of loss and recovery of boundary conditions, obtained in Theorems 1.1–1.4 below have an interesting interpretation: when the system (controlled Brownian particle) starts near the boundary, for certain spatial distributions of rewards inside the domain, the maximal gain attains profitable values on different time horizons but not on some intermediate times.

1.3. Results in general domains

Our first main result in general domains is the following. It shows the existence of solutions undergoing GBU with losses and recoveries of boundary conditions at arbitrarily many times.

Theorem 1.1.
Let $p > 2$ and let Ω be any smoothly bounded domain of $R^{n}$ . For any integer $m ⩾ 2$ , there exists $ϕ \in X$ such that the corresponding solution undergoes GBU with at least m losses and recoveries of boundary conditions. More precisely, we have $T (ϕ) < \infty$ and there exist times $s_{m} > \dots > s_{1} > T (ϕ)$ and nonempty open subintervals $J_{i} \subset (s_{i}, s_{i + 1})$ for $i = 1, \dots, m - 1$ , such that $\begin{array}{l} (1.2) & sup_{x \in \partial Ω} u (x, s_{i}) > 0 for each i \in {1, \dots, m}, \\ (1.3) & \begin{aligned} u is a classical solution on J_{i} for each i \in {1, \dots, m - 1}, \\ i.e., u \in C^{2, 1} (\overline{Ω} \times J_{i}) and u = 0 on \partial Ω \times J_{i} . \end{aligned} \end{array}$

We can also show that the scenario in Theorem 1.1 can be preceded by the first GBU without LBC. Theorem 1.2.
Let $p > 2$ and let Ω be any smoothly bounded domain of $R^{n}$ . For any integer $m ⩾ 2$ , there exists $ϕ \in X$ such that u undergoes:

∙ GBU without LBC at $t = T (ϕ) < \infty$ , i.e. $u = 0$ on $\partial Ω \times [0, T (ϕ) + τ]$ for some $τ > 0$ ;

∙ and then at least m losses and recoveries of boundary conditions; namely there exist times $s_{m} > \dots > s_{1} > T (ϕ) + τ$ and nonempty open subintervals $J_{i} \subset (s_{i}, s_{i + 1})$ for $i = 1, \dots, m - 1$ , such that ( 1.2 )–( 1.3 ) holds.

1.4. Results in one space dimension

We will obtain more precise results in the case of $n = 1$ . We only consider the behavior at $x = 0$ . Similar results at $x = 1$ immediately follow by considering the reflected solution $\tilde{u} (x, t) = u (1 - x, t)$ .

Set $\begin{array}{l} X_{1} : = {ϕ \in C^{1} ([0, 1]); ϕ ⩾ 0, ϕ (0) = ϕ (1) = 0}, \end{array}$ denote $\begin{array}{l} U^{*} (x) = c_{p} x^{α}, x > 0, with α = \frac{p - 2}{p - 1}, c_{p} = {(p - 2)}^{- 1} {(p - 1)}^{(p - 2) / (p - 1)}, \end{array}$ the singular steady state vanishing at $x = 0$ , and set $\begin{array}{l} (1.4) & N (t) : = z (u (\cdot, t) - U^{*}), \end{array}$ where z denotes the number of sign changes on $[0, 1]$ , i.e. $\begin{array}{l} z (v) = sup {m \in N; \exists x_{0} < \dots < x_{m} \in (0, 1), v (x_{i - 1}) v (x_{i}) < 0, i = 1, \dots, m}, \end{array}$ with the convention $z (v) = - \infty$ in case $v \equiv 0$ . Let us also recall (see [34, Lemma 5.1]) that $\begin{array}{l} (1.5) & u_{x} (0, t) = lim_{x \to 0} u_{x} (x, t) \in R \cup {+ \infty}, t > 0 \end{array}$ (where the limit exists, possibly $+ \infty$ ). A crucial ingredient in our analysis is the study of the following “transition” set: $\begin{array}{l} (1.6) & T = T_{ϕ} : = {t > 0; u (0, t) = 0 and u_{x} (0, t) = \infty}, \end{array}$ in connection with the intersection number properties of the solution u with the singular steady state $U^{*}$ . For future reference, we also denote the singular set $\begin{array}{l} (1.7) & S = S_{ϕ} : = {t > 0; u_{x} (0, t) = \infty} . \end{array}$

The following two theorems are our main one-dimensional results. The first one gives a classification of all possible behaviors at any time, for any viscosity solution to problem (1.1).

Theorem 1.3.
Let $p > 2$ , $n = 1$ , $Ω = (0, 1)$ , $ϕ \in X_{1}$ .

(i) The set $T = T_{ϕ}$ is at most finite. Moreover, if $N (0) < \infty$ then we have $\begin{array}{l} (1.8) & # T ⩽ N (0) . \end{array}$

(ii) On each interval between two consecutive times $t_{1}, t_{2} \in T$ , the solution is either:

∙ classical up to $x = 0$ , i.e.: $\begin{array}{l} (1.9) & u \in C^{2, 1} ([0, \frac{1}{2}] \times (t_{1}, t_{2})) and u = 0 on {0} \times (t_{1}, t_{2}) \end{array}$

∙ or of LBC type at $x = 0$ , i.e.; $\begin{array}{l} (1.10) & u > 0 on {0} \times (t_{1}, t_{2}) . \end{array}$

Our results also answer another question left open in [34]: as a consequence of Theorem 1.3, we see that waiting time phenomena cannot occur, at least in one space dimension. Once a solution undergoes gradient blowup, the solution either loses boundary conditions immediately or is immediately regularized.

The next result is in some sense the reciprocal of Theorem 1.3. It shows that any given finite sequence of behaviors a priori permitted by Theorem 1.3 is actually realized for suitable choice of initial data. In the following, the letters $C, L$ respectively stand for “Classical up to $x = 0$ ” and “LBC at $x = 0$ ” (cf. (1.9)–(1.10)).
Theorem 1.4.
Let $p > 2$ , $n = 1$ , $Ω = (0, 1)$ and let ℓ be a positive integer. Let ${(σ_{i})}_{0 ⩽ i ⩽ ℓ}$ be any finite sequence with values in ${C, L}$ , such that $σ_{0} = σ_{ℓ} = C$ , and set $t_{0} = 0$ and $t_{ℓ + 1} = \infty$ . There exist $ϕ \in X_{1}$ and times $0 < t_{1} < \dots < t_{ℓ} \in T$ such that
for each $i \in {0, \dots, ℓ}$ , the behavior of u on $(t_{i}, t_{i + 1})$ is of type $σ_{i}$

u is classical up to $x = 1$ for all times, i.e. $u \in C^{2, 1} ([\frac{1}{2}, 1] \times (0, \infty))$ and $u = 0$ on ${1} \times (0, \infty)$ .

Typical behaviors given in Theorems 1.3 and 1.4 are illustrated in Fig. 1–3.

The name transition set for $T$ is motivated by the fact that, in view of Theorems 1.3 and 1.4, the elements of $T$ play the role of transition times between intervals with behaviors C, L. The passage from classical to LBC (resp., to classical) is associated with GBU with (resp., without) LBC. As for the passage from LBC to classical it corresponds to the phenomenon of regularization. Here we discover a new type of behavior, which we call bouncing. This is when the solution passes from LBC to LBC, through a single time of recovery of boundary conditions. Whereas solutions with separated LBC time intervals should be rather stable, the phenomena of GBU without LBC or of bouncing are expected to be unstable.

Fig. 1.
A solution with exactly 4 losses and recoveries of boundary conditions.

Fig. 2.
A solution with exactly 1 bouncing.

Fig. 3.
A solution with mixed behaviors (2 LBC, 2 GBU without LBC and 1 double bouncing).
Remark 1.1 (Zero number properties).

(i) Assume $\begin{array}{l} (1.11) & ‖ ϕ ‖_{\infty} ⩽ c_{p} \end{array}$ or ϕ symmetric and let $t \in T$ . It follows from Proposition 7.2 below that, if the solution changes type through time t, then it must “lose an odd number of intersections with $U^{*}$ ”, more precisely, ${lim}_{s \to t^{-}} N (s) - N (t)$ is odd. On the other hand, if it does not change type, then it must lose an even number of intersections with $U^{*}$ . More generally this remains true provided $u (1, t) \neq c_{p}$ . It $u (1, t) = c_{p}$ , then the above remains true if we replace N with the number of sign changes of $u (\cdot, t) - U^{*}$ over $[0, b]$ with $b < 1$ close enough to 1.

Remark 1.2 (Radial case in higher dimensions).

Consider (1.1) in the case that Ω is a ball or an annulus in $R^{n}$ . Let $Ω = (ρ, 1)$ with $ρ \in (0, 1)$ . Then (1.1) for a radial solution turns out $\begin{array}{l} (1.12) & \{\begin{matrix} u_{t} - u_{r r} - \frac{n - 1}{r} u_{r} = | u_{r} |^{p}, & ρ < r < 1, t > 0, \\ u = 0, & r \in {ρ, 1}, t > 0, \\ u (r, 0) = ϕ (r), & ρ < r < 1, \end{matrix} \end{array}$ with $r : = | x |$ . The singular steady state with singularity at $r = 1$ is given by $\begin{array}{l} (1.13) & U_{rad}^{*} (r) = \int_{r}^{1} ξ^{1 - n} {[\frac{(p - 1) (ξ^{1 - (n - 1) (p - 1)} - 1)}{(n - 1) (p - 1) - 1}]}^{- \frac{1}{p - 1}} d ξ, ρ < r < 1 . \end{array}$ It is immediate that $U_{rad}^{*}$ behaves similarly to $U^{*}$ near $r = 1$ , i.e., $\begin{array}{l} U_{rad}^{*} (r) = c_{n, p} {(1 - r)}^{α} + o ({(1 - r)}^{α}) for r < 1 close to 1 \end{array}$ with some $c_{n, p} > 0$ . The proofs of Theorems 1.3 and 1.4 can be easily modified using $U_{rad}^{*}$ instead of $U^{*}$ . Similar modifications work in the case $Ω = B_{1}$ , provided we assume that $‖ ϕ ‖_{\infty} ⩽ U_{rad}^{*} (0)$ , given by formula (1.13).

Remark 1.3 (Further development).

Our solutions given in Theorem 1.4 lose at most two intersections with $U^{*}$ at each $t \in T$ . There should be other possibilities. In addition, we do not deal with GBU rate and recovery rate at $t \in T$ in this paper. If these questions are solved, then the behavior of global viscosity solutions will be completely clarified. We used blow-up profile given in [34] in the proofs of Theorems 1.3 and 1.4. However we could replace them by zero number arguments. These together with the iterative method based on the continuity of critical parameters in the proof of Theorem 1.4 will play a crucial role. This kind of methods will be applied to other equations and we will treat these issues in forthcoming papers.

1.5. Differences from Fujita equation

Blow-up problems in nonlinear parabolic equations have been studied the most extensively for the so-called Fujita equation $\begin{array}{l} (1.14) & \{\begin{matrix} u_{t} - Δ u = u^{q}, & x \in R^{n}, t > 0, \\ u (x, 0) = ϕ (x) ⩾ 0, & x \in R^{n} \end{matrix} \end{array}$ with $q > 1$ . A solution u of (1.14) is said to blow up at $t = T < \infty$ if ${lim sup}_{t \to T} ‖ u (t) ‖_{L^{\infty}} = \infty$ . There are critical values of the exponent q related to continuation after blow-up time in (1.14). Denote by $q_{S} = (n + 2) / {(n - 2)}_{+}$ the Sobolev exponent and by $q_{J L}$ the Joseph–Lundgren exponent, defined by $\begin{array}{l} (1.15) & q_{J L} = \{\begin{matrix} + \infty & if n ⩽ 10, \\ 1 + \frac{4}{n - 4 - 2 \sqrt{n - 1}} & if n ⩾ 11. \end{matrix} \end{array}$ When a solution u of (1.14) blows up at $t = T$ , the blow-up is called complete if the proper continuation for $t > T$ identically equals $+ \infty$ in $R^{n} \times (T, \infty)$ , and incomplete otherwise. We refer to [20] for the definition of proper solution and its main properties. Roughly speaking, a proper solution is a weak solution defined as a limit of global classical solutions to an approximate equation. From the point of view of regularization after blow-up, complete and incomplete blow-up in (1.14) correspond to GBU with LBC and without LBC in (1.1), respectively. In the case $q < q_{S}$ , only the complete blowup is possible by [6]. On the other hand, besides complete blow-up solutions, incomplete blow-up occurs when $q ⩾ q_{S}$ [26,31]. We note that these works do not give information on the behavior of weak solutions after blow-up time. The simplest incomplete blow-up solution is a peaking solution and its existence is known from [20,27] (see also [16,25]). Here a global weak solution u is called a peaking solution if u blows up at $t = T < \infty$ , recovers regularity just after $t = T$ and exists for $t > T$ in the classical sense. The GBU solution of (1.1) without LBC constructed in [34] is similar to a peaking solution of (1.14). The natural question on the existence of a solution with multiple blow-up times has been paid attention in the field of the semilinear heat equation and was answered affirmatively for $q > q_{J L}$ in [28,29]. Namely, if $q > q_{J L}$ , then for arbitrary $m \in N^{*}$ there exists a radial solution u of (1.14) and $0 < t_{1} < t_{2} < \dots < t_{m}$ such that u blows up at $t = t_{i}$ and that u is classical in $(t_{i - 1}, t_{i})$ for $i = 1, 2, \dots, m$ , where $t_{0} : = 0$ . The radial solution with multiple blow-up times was constructed in [28,29] by carefully choosing initial data with m isolated parts such that each part corresponds to initial data of the special radial solution by [22] blowing up around $t = t_{i}$ for $i = 1, 2, \dots, m$ . While the solutions due to [28,29] blow up only at the origin at each blow-up time, a solution of different type, which undergoes incomplete blow-up at the origin at the first blow-up time and complete blow-up on a sphere at the second blow-up time, was constructed in [30].

Whereas complete blow-up solutions of the Fujita equation (1.14) cease to exist in any weak sense at blow-up time, solutions of (1.1) undergoing GBU with LBC continue to exist in viscosity sense after blow-up time and recover regularity after a while. This makes the description of behavior of viscosity solutions after blow-up time more complicated than in (1.14). In other words, when a radial solution of (1.14) undergoes blow-up at multiple blow-up times, only the immediate regularization is possible except at final blow-up time. On the contrary, various combinations of GBU with LBC and without LBC turn out to occur in (1.1), including the new type of behavior called bouncing. Accordingly, as we shall explain in the next subsection, we introduce a method based on an arbitrary number of critical parameters, whose continuity requires a delicate argument. Since we do not rely on any known special solution unlike in the proof for Fujita equation, our method is expected to apply to other equations.

1.6. Ideas of proofs

(i) The main idea of the proof of Theorem 1.1 is as follows. We construct a multiscale, compactly supported initial data, made of m, suitably rescaled, bumps which are located farther and farther from the boundary. The distribution of the sizes and locations of the bumps in terms of the distance to the boundary is rather delicate and the construction has to be made in a recursive way. The bump which is closest to the boundary generates the first GBU and LBC. The second bump is much farther from the boundary and its influence becomes significant only after some lapse of time, leaving enough time for the solution to get regularized by an effect of the diffusion, before producing a second GBU and LBC. Repeating the process, we construct an m-bump initial data which leads to a solution undergoing GBU and LBC (at least) m times.1

¹
Although the above description is more convenient for heuristic purposes, the actual construction is done in the converse direction, first constructing the bump which is farthest from the boundary.

The estimates leading to each LBC and regularization are provided by suitable rescaling and comparison arguments, partly inspired by [24] and [35], that are the object of the preliminary lemmas in Section 3. As for the proof of Theorem 1.2, it is based on a modification of the proof of Theorem 1.1 and a limiting argument.

(ii) The basic idea of the proof of Theorem 1.3(i) is to show that $N (t)$ is a nondecreasing function for $t ⩾ 0$ and that $N (t)$ has to drop at any transition time $t \in T$ . These properties can be shown to be true provided one makes the additional assumption (1.11) (see Propositions 6.1 and 7.2) and this then readily yields the bound (1.8). In the general case, without assumption (1.11), $N (t)$ need not be monotone (see Proposition 6.3) and the proof requires additional arguments. In particular, one needs to study the intersection properties of approximate solutions $u_{k}$ with regular steady states $U_{j}$ for suitable large $j, k$ . As for the proof of Theorem 1.3(ii) it is based on a further analysis of intersection properties of the solution with $U^{*}$ , combined with asymptotic estimates of the boundary profile of the solution at times $t \in S$ .

(iii) The proof of Theorem 1.4 is rather long and delicate. It is based on a modification of the multibump construction in the proof of Theorems 1.1 combined with deformation, zero number and recursion arguments. More precisely we construct a multi-parameter family of initial data based on suitable deformations of multibumps initial data, and the desired solution is obtained by iteratively selecting appropriate critical values of the parameters. The required continuity properties of the critical parameter functions rely upon zero number arguments applied to the difference of two solutions, whereas the exact structure of the resulting solution depends on dropping properties of the number of intersections with $U^{*}$ . In the construction stated in (i), the behavior of solutions corresponding to each bump of initial data is rather stable, and does not give a serious effect to the others. On the other hand, the behaviors associated with the critical parameters are unstable and sensitive to the other parts. That is a reason why the proof is long and complicated.

2. Preliminary results I: Approximation, comparison and regularity

Throughout the paper, we shall denote the function distance to the boundary by $\begin{array}{l} δ (x) = dist (x, \partial Ω), x \in Ω, \end{array}$ and by ν the outer normal vector to $\partial Ω$ .

We start by recalling (see, e.g., [34,35]) that (1.1) can be approximated (away from the boundary) by the truncated problems $\begin{array}{l} (2.1) & \{\begin{matrix} \partial_{t} u_{k} - Δ u_{k} = F_{k} (| \nabla u_{k} |^{2}), & x \in Ω, t > 0, \\ u_{k} = 0, & x \in \partial Ω, t > 0, \\ u_{k} (x, 0) = ϕ (x), & x \in Ω, \end{matrix} \end{array}$ where $\begin{array}{l} (2.2) & F_{k} (s) = min (s^{p / 2}, k^{(p - 2) / 2} s) . \end{array}$ Namely, (2.1) admits a unique, global classical solution $u_{k} \in C^{1, 2} (\overline{Ω} \times (0, \infty))$ with $u_{k}, \nabla u_{k} \in C (\overline{Ω} \times [0, \infty))$ , the sequence $u_{k}$ is nondecreasing, and $\begin{array}{l} u_{k} \to u in C_{loc}^{2, 1} (Ω \times (0, \infty)), as k \to \infty . \end{array}$

We next give two versions of the comparison principle that will be used repeatedly in what follows. The first one is a standard comparison principle for sub-/supersolutions (see e.g. [40, Proposition 2.1])

Proposition 2.1.
Let ω be any bounded open subset of $R^{n}$ . Let $0 ⩽ t_{1} < t_{2}$ , set $Q : = ω \times (t_{1}, t_{2})$ and denote by $\partial_{P} Q$ its parabolic boundary. Assume that $u_{i} \in C^{2, 1} (Q) \cap C (\overline{Q})$ , $i = 1, 2$ , satisfy $\begin{array}{l} \partial_{t} u_{1} - Δ u_{1} - | \nabla u_{1} |^{p} ⩽ \partial_{t} u_{2} - Δ u_{2} - | \nabla u_{2} |^{p} in Q_{T}, \end{array}$ Then $\begin{array}{l} sup_{Q} (u_{1} - u_{2}) ⩽ sup_{\partial_{P} Q} (u_{1} - u_{2}) . \end{array}$

The second one is a suitable form of the comparison principle for viscosity solutions. A key point is that comparison is guaranteed although we only assume $w ⩾ 0$ (in the classical sense) on $\partial Ω$ for the comparison function from above, whereas the viscosity solution u may take positive pointwise values on the boundary (but of course it satisfies $u = 0$ in the generalized viscosity sense or, equivalently, in the sense of monotone approximation by classical solutions of truncated problems).
Proposition 2.2.
Let $ϕ \in X$ and let u be the corresponding global viscosity solution of ( 1.1 ). Let ω be any open subset of Ω, let $t_{2} > t_{1} ⩾ 0$ and set $Q : = ω \times (t_{1}, t_{2})$ . Assume that $w \in C^{1, 2} (Q) \cap C (\overline{Q})$ satisfies $\begin{array}{l} w_{t} - Δ w ⩾ | \nabla w |^{p} in Q, \\ w ⩾ 0 on (ω \cap \partial Ω) \times (t_{1}, t_{2}), \\ w ⩾ u on (Ω \cap \partial ω) \times (t_{1}, t_{2}), \\ w (\cdot, t_{1}) ⩾ u (\cdot, t_{1}) in ω . \end{array}$ Then $w ⩾ u$ in $\overline{Q}$ .

For convenience, we give a short proof.
Proof.
Consider the smooth solutions $u_{k}$ of the truncated problems (2.1). Since $u ⩾ u_{k}$ , our assumptions imply that $w ⩾ u_{k}$ on the parabolic boundary of Q. Since $F_{k} (| \nabla v |^{2}) ⩽ | \nabla v |^{p}$ , it follows from Proposition 2.2 that $w ⩾ u_{k}$ on Q. For each $(x, t) \in Q$ , passing to the limit $k \to \infty$ , we deduce that $w (x, t) ⩾ u (x, t)$ . Finally, since $u, w \in C (\overline{Ω} \times [0, \infty))$ , this extends to $\overline{Q}$ . □

The solution u satisfies the following continuous dependence estimate with respect initial data (see e.g. [34, Theorem 3.1]): $\begin{array}{l} (2.3) & {‖ u (ϕ_{1}; t) - u (ϕ_{2}; t) ‖}_{\infty} ⩽ ‖ ϕ_{1} - ϕ_{2} ‖_{\infty}, for all t > 0 . \end{array}$ As another useful maximum principle property, let us recall (see, e.g., [33, Lemma 3.1] and [34, Section 3]) that for all $t_{0} > 0$ , there exist $M = M (t_{0}) > 0$ and $k_{0} ⩾ 1$ such that $\begin{array}{l} (2.4) & | u_{t} | ⩽ M in Ω \times [t_{0}, \infty) \end{array}$ and the solutions $u_{k}$ of the truncated problems (2.1) satisfy $\begin{array}{l} (2.5) & | \partial_{t} u_{k} | ⩽ M in Ω \times [t_{0}, \infty) for all k ⩾ k_{0} . \end{array}$ Since $u \in C (\overline{Ω} \times [0, \infty))$ , (2.4) guarantees the following global Lipschitz estimate in time up to the boundary $\begin{array}{l} (2.6) & | u (x, t) - u (x, s) | ⩽ M (t - s), x \in \overline{Ω}, t_{0} ⩽ s < t < \infty . \end{array}$

We next recall (see [8]) that the existence of a unique global viscosity solution u of (1.1), with $u \in C^{1, 2} (\overline{Ω} \times (0, T)) \cap C (\overline{Ω} \times [0, \infty))$ , remains true for any initial data $ϕ \in C_{0} (Ω)$ . However, unlike in the case of $C^{1}$ initial data (namely $u_{0} \in X$ ), u need then not be smooth up to the boundary, nor achieve the boundary conditions in the classical sense, even for small time. We first give the following proposition, which guarantees classical regularity for small time provided the initial data $ϕ \in C_{0} (Ω)$ is merely controlled above by the distance to the boundary, even though ϕ is not $C^{1}$ . This result will be a useful tool in the proof of Theorem 1.1, to ensure that the solution becomes classical again between two losses of boundary conditions. Set $\begin{array}{l} Y : = {ϕ \in C (\overline{Ω}); ϕ = 0 on \partial Ω and {sup}_{Ω} \frac{ϕ}{δ} < \infty} . \end{array}$
Proposition 2.3.
Let $ϕ \in Y$ . There exists $τ > 0$ such that problem ( 1.1 ) admits a unique classical solution U on $[0, τ]$ , with $U \in C^{1, 2} (\overline{Ω} \times (0, τ]) \cap C (\overline{Ω} \times [0, τ])$ . Moreover, we have $\begin{array}{l} (2.7) & sup_{t \in (0, τ]} t^{1 / 2} {‖ \nabla U (t) ‖}_{\infty} < \infty, \end{array}$ and U coincides with the unique viscosity solution u on $[0, τ]$ .
Remark 2.1.
The time τ can be chosen uniform in terms of $\begin{array}{l} (2.8) & M : = max {sup_{Ω} δ^{- 1} ϕ, | inf_{Ω} ϕ |} . \end{array}$

Altough Proposition 2.3 does not seem to have appeared in the literature, it is essentially a consequence of Bernstein-type and approximation arguments from [35]. Since the motivation in [35] was ultimate regularization for large time, we need to adapt them to our present purpose and we give a full proof for completeness. Proof of Proposition 2.3.
Let $ψ \in C^{2} (\overline{Ω})$ be the unique solution of the linear problem $\begin{array}{l} (2.9) & \{\begin{matrix} - Δ ψ = 1, & x \in Ω, \\ ψ = 0, & x \in \partial Ω . \end{matrix} \end{array}$ By Hopf’s lemma, there exists $c_{1} > 0$ such that $\begin{array}{l} (2.10) & ψ ⩾ c_{1} δ in Ω . \end{array}$ By assumption, we have $ϕ ⩽ M δ ⩽ c_{1} M ψ$ , where M is given by (2.8). Let w be the solution of problem (1.1) with initial data $c_{1} M ψ \in X$ . There exist $τ > 0$ and $C_{1} > 0$ depending only on $M, p, Ω$ , such that w is classical on $(0, τ]$ and $\begin{array}{l} (2.11) & sup_{t \in (0, τ]} {‖ \nabla w (t) ‖}_{\infty} ⩽ C_{1} . \end{array}$

As in the proof of Proposition 2.2, we next introduce a sequence of truncated problems, this time with slightly better behaved truncated nonlinearities. For each integer $k ⩾ 1$ , let $F_{k}$ be given by $\begin{array}{l} (2.12) & F_{k} (s) = \{\begin{matrix} s^{p / 2}, & 0 ⩽ s ⩽ k, \\ k^{p / 2} + \frac{p}{2} k^{(p - 2) / 2} (s - k), & s > k . \end{matrix} \end{array}$ Note that $F_{k} \in C^{1} ([0, \infty))$ , with $F_{k}^{'}$ locally Hölder continuous, $\begin{array}{l} (2.13) & 0 ⩽ F_{k} (s) ⩽ | s |^{p / 2}, s ⩾ 0 \end{array}$ and $\begin{array}{l} (2.14) & 2 s F^{'} (s) ⩾ F (s), s ⩾ 0 . \end{array}$ Since $F_{k} (| \nabla v |^{2})$ has at most quadratic growth at infinity with respect to $\nabla v$ , it is well known that problem (2.1) admits a unique, global classical solution $v = v_{k}$ . By the maximum principle, we have $\begin{array}{l} (2.15) & | v | ⩽ ‖ ϕ ‖_{\infty} \end{array}$ and, in view of (2.13), it follows from the comparison principle that $\begin{array}{l} v ⩽ w in Q : = Ω \times (0, τ) . \end{array}$ Consequently, using (2.11), we get $\begin{array}{l} (2.16) & \frac{\partial v}{\partial ν} ⩾ - C_{1} in Q . \end{array}$ On the other hand, we have $v ⩾ z : = e^{t Δ} (ϕ_{-})$ , where $ϕ_{-} = min (ϕ, 0)$ . Since, by standard heat semigroup estimate, $‖ \nabla z (t) ‖_{\infty} ⩽ C (Ω) t^{- 1 / 2} ‖ ϕ_{-} ‖_{\infty} ⩽ C (Ω) M t^{- 1 / 2}$ , we deduce that $\begin{array}{l} (2.17) & \frac{\partial v}{\partial ν} ⩽ C (Ω) t^{- 1 / 2} M in Q . \end{array}$

Now, following [35], in order to obtain gradient estimates for $v = v_{k}$ away from $t = 0$ , uniformly in k, we introduce the following Bernstein-type auxiliary function: $\begin{array}{l} J = t | \nabla v |^{2} + {(v + ‖ ϕ ‖_{\infty})}^{2} . \end{array}$ By (2.15)–(2.17), we have $\begin{array}{l} J ⩽ C_{2} : = 4 ‖ ϕ ‖_{\infty}^{2} + max (C^{2} (Ω) M^{2}, C_{1}^{2} τ) on \partial_{P} Q \end{array}$ (where $\partial_{P}$ denotes the parabolic boundary). We compute, in Q: $\begin{array}{l} J_{t} = | \nabla v |^{2} + 2 t \nabla v \cdot \nabla v_{t} + 2 (v + ‖ ϕ ‖_{\infty}) v_{t} \end{array}$ and, using $Δ (| \nabla v |^{2}) = 2 \nabla v \cdot \nabla Δ v + 2 \sum_{i j} {(v_{i j})}^{2}$ , $\begin{array}{l} Δ J ⩾ 2 t \nabla v \cdot \nabla Δ v + 2 | \nabla v |^{2} + 2 (v + ‖ ϕ ‖_{\infty}) Δ v . \end{array}$ Therefore, $\begin{array}{l} J_{t} - Δ J & ⩽ - | \nabla v |^{2} + 2 t \nabla v \cdot \nabla (F (| \nabla v |^{2})) + 2 (v + ‖ ϕ ‖_{\infty}) F (| \nabla v |^{2}) . \end{array}$ Since $\begin{array}{l} 2 t \nabla v \cdot \nabla (F (| \nabla v |^{2})) & = 4 t \sum_{i} v_{i} \nabla v \cdot \nabla v_{i} F^{'} (| \nabla v |^{2}) = 2 \nabla v \cdot \nabla (t | \nabla v |^{2}) F^{'} (| \nabla v |^{2}) \\ = 2 F^{'} (| \nabla v |^{2}) \nabla v \cdot \nabla J - 4 (v + ‖ ϕ ‖_{\infty}) | \nabla v |^{2} F^{'} (| \nabla v |^{2}), \end{array}$ it follows that $\begin{array}{l} J_{t} - Δ J - 2 F^{'} (| \nabla v |^{2}) \nabla v \cdot \nabla J ⩽ - | \nabla v |^{2} + 2 (v + ‖ ϕ ‖_{\infty}) [F (| \nabla v |^{2}) - 2 | \nabla v |^{2} F^{'} (| \nabla v |^{2})] . \end{array}$ Using (2.14), (2.15), we deduce that $\begin{array}{l} J_{t} - Δ J - 2 F^{'} (| \nabla v |^{2}) \nabla v \cdot \nabla J ⩽ 0 . \end{array}$ It follows from the maximum principle that $\begin{array}{l} (2.18) & sup_{Q} J ⩽ sup_{\partial_{P} Q} J ⩽ C_{2} . \end{array}$ Since $C_{2}$ is independent of k, this guarantees that the sequence $(\nabla v_{k})$ is uniformly bounded on $Q_{ε} : = Ω \times [ε, τ]$ for each $ε \in (0, τ)$ , and it follows from parabolic estimates that $(v_{k})$ is relatively compact in $C^{1, 2} (\overline{Q_{ε}})$ . By a diagonal procedure, we deduce that some subsequence of $v_{k}$ converges to a classical solution U of (1.1)₁–(1.1)₂ in $\overline{Ω} \times (0, τ]$ , with convergence in $C^{1, 2} (\overline{Q_{ε}})$ for each $ε \in (0, τ)$ . Moreover, for each $t \in (0, τ]$ , (2.18) implies $t ‖ \nabla U (\cdot, t) ‖_{\infty}^{2} ⩽ {lim sup}_{k} t ‖ \nabla v_{k} (\cdot, t) ‖_{\infty}^{2} ⩽ C_{2}$ , hence estimate (2.7).

It remains to show that $\begin{array}{l} (2.19) & U \in C (\overline{Ω} \times [0, τ]) with U (0) = ϕ . \end{array}$ Since $ϕ \in C_{0} (Ω)$ , for each $η > 0$ , we may pick a function $ψ_{η} \in C_{0}^{\infty} (Ω)$ such that $‖ ϕ - ψ_{η} ‖_{\infty} ⩽ η$ . Setting $K = {sup}_{Ω} (Δ ψ_{η} + | \nabla ψ_{η} |^{q})$ and $\overline{v} (x, t) = ψ_{η} + η + K t$ , and using (2.13), we have $\begin{array}{l} {\overline{v}}_{t} - Δ \overline{v} - F_{k} (| \nabla \overline{v} |^{2}) = K - Δ ψ_{η} - F_{k} (| \nabla ψ_{η} |^{2}) ⩾ K - Δ ψ_{η} - | \nabla ψ_{η} |^{p} ⩾ 0 . \end{array}$ Since $\overline{v} (x, 0) = ψ_{η} + η ⩾ ϕ$ , it follows from the comparison principle that $\begin{array}{l} e^{t Δ} ϕ ⩽ v_{k} ⩽ ψ_{η} + η + K t ⩽ ϕ + 2 η + K t . \end{array}$ Consequently, $‖ v_{k} (\cdot, t) - ϕ ‖_{\infty} ⩽ 3 η$ for $t > 0$ small, hence (2.19).

Finally, since U is in particular a viscosity solution, it has to coincide with the unique viscosity solution u. This in turn guarantees the uniqueness of U. The proof of Proposition 2.3 is complete. □

3. Preliminary results II: LBC and regularity properties in time based on local properties of initial data

For each $ε > 0$ , we set $\begin{array}{l} Ω_{ε} = {x \in Ω; δ (x) < ε}, Ω^{ε} = {x \in Ω; δ (x) > ε}, Γ_{ε} = {x \in Ω; δ (x) = ε} \end{array}$ (observing that $Ω_{ε} = Ω$ and $Ω^{ε}, Γ_{ε} = \emptyset$ for $ε > 0$ sufficiently large). The goal of this section is to establish the following two propositions, which are important ingredients for the proofs of Theorems 1.1, 1.2 and 1.4.

In our first proposition, we show that LBC in short time occurs for suitable initial data, with support concentrated near the boundary and precisely controlled, small $L^{\infty}$ norm. We also give a useful, slightly more precise version in the case $n = 1$ .

Proposition 3.1.
(i) There exists $ε_{0} \in (0, 1)$ such that, for all $ε \in (0, ε_{0})$ , there exist a compactly supported $ψ_{ε} \in X$ with the following properties: $\begin{array}{l} (3.1) & Supp (ψ_{ε}) \subset Ω_{2 ε} ∖ Ω_{ε} \\ (3.2) & ‖ ψ_{ε} ‖_{\infty} ⩽ K_{1} ε^{α}, \\ (3.3) & T (ψ_{ε}) < τ_{ε} = c_{0} ε^{2}, \\ (3.4) & sup_{x \in \partial Ω} u (ψ_{ε}; x, τ_{ε}) > 0, \end{array}$ where $K_{1}, c_{0} > 0$ are constants independent of ε.

(ii) Let $n = 1$ , $Ω = (0, 1)$ . There exist $c_{2} > c_{1} > 0$ and $C_{0} = C_{0} (p) > 0$ and, for each $η \in (0, 1)$ , there exists $\tilde{K} = \tilde{K} (η) > 0$ such that, for any $ε \in (0, 1 / 2)$ , if $\begin{array}{l} (3.5) & ϕ ⩾ \tilde{K} ε^{α} on ((1 - η) ε, (1 + η) ε), \end{array}$ then $\begin{array}{l} (3.6) & u (0, t) ⩾ C_{0} ε^{α} for all t \in [c_{1} ε^{2}, c_{2} ε^{2}] . \end{array}$

Our second proposition gives a regularity property for the solution at suitable times, depending on the behavior of the initial data at given distance from the boundary.
Proposition 3.2.
Let $ϕ \in X \cap C^{2} (Ω)$ . There exists a constant $L = L (Ω, p) > 0$ and for each $ε, M > 0$ , there exists a constant $γ_{0} = γ_{0} (Ω, p, ε, M) > 0$ with the following property. If for some function $H \in C^{2} (\overline{Ω})$ , $H ⩾ 0$ , such that $\begin{array}{l} (3.7) & ‖ H ‖_{C^{2} ({\overline{Ω}}^{ε / 2})} ⩽ M \end{array}$ and some constant $γ \in [0, γ_{0})$ we have $\begin{array}{l} (3.8) & ϕ ⩽ \{\begin{matrix} H & in Ω^{ε}, \\ γ & in Ω_{ε}, \end{matrix} \end{array}$ then u is a classical solution for $t \in [L γ, L γ_{0}]$ .
Proof of Proposition 3.1.
Consider the following problem $\begin{array}{l} (3.9) & \{\begin{matrix} v_{t} - Δ v = | \nabla v |^{p}, & x \in B_{1}, t > 0, \\ v = 0, & x \in \partial B_{1}, t > 0, \\ v (x, 0) = v_{0} (x), & x \in B_{1}, \end{matrix} \end{array}$ where $0 ⩽ v_{0} \in C_{0}^{\infty} (R^{n})$ is radially symmetric with $Supp (v_{0}) \subset B_{1 / 4}$ . Denote by v the unique global viscosity solution of (3.9) and by $T_{0} : = T (v_{0})$ its maximal existence time as a classical solution. After replacing $v_{0}$ by a sufficiently large multiple, it follows from [33, Theorem 4] that $T_{0} < \infty$ and that there exists $t_{0} > T_{0}$ such that $\begin{array}{l} (3.10) & sup_{x \in \partial B_{1}} v (x, t_{0}) > 0 . \end{array}$

Next pick some point $a \in Ω$ , let b be its projection onto $\partial Ω$ , set $d = | a - b |$ and let $e : = d^{- 1} (a - b)$ be the inward unit normal vector to $\partial Ω$ at the point b. For each $ε \in (0, d)$ , we set $a_{ε} = b + ε e$ . Then $D_{ε} : = B (a_{ε}, ε) \subset Ω$ and $b \in \partial Ω$ .

Now, following [24], we consider the following rescaled functions: $\begin{array}{l} (3.11) & ϕ_{ε} (x) : = ε^{α} v_{0} (ε^{- 1} (x - a_{ε})) ⩾ 0, x \in R^{n} \end{array}$ (which is supported in $D_{ε}$ ) and $\begin{array}{l} v_{ε} (x, t) = ε^{α} v (ε^{- 1} (x - a_{ε}), ε^{- 2} t), (x, t) \in {\overline{D}}_{ε} \times (0, \infty] . \end{array}$ Then $v_{ε}$ solves problem (1.1) with Ω replaced by $B (a_{ε}, ε)$ and initial data $ϕ_{ε}$ . Let $u = u (ϕ_{ε}; \cdot, \cdot)$ be the solution of problem (1.1) (in Ω) with initial data $ϕ = ϕ_{ε}$ . By Proposition 2.2, using $u ⩾ 0$ , we get $\begin{array}{l} u ⩾ v_{ε} in D_{ε} \times (0, \infty) . \end{array}$ Since v is radially symmetric, it follows from (3.10) that $\begin{array}{l} (3.12) & u (ϕ_{ε}; b, ε^{2} t_{0}) ⩾ v_{ε} (b, ε^{2} t_{0}) = ε^{α} v (ε^{- 1} (b - a_{ε}), t_{0}) = ε^{α} v (- e, t_{0}) > 0 . \end{array}$ On the other hand, we note that $\begin{array}{l} (3.13) & \begin{aligned} ϕ_{ε} (x) \neq 0 & ⟹ | ε^{- 1} (x - a_{ε}) | < 1 / 4 \\ ⟹ | δ (x) - ε | = | δ (x) - δ (a_{ε}) | ⩽ | x - a_{ε} | < ε / 4 \\ ⟹ \frac{3 ε}{4} < δ (x) < \frac{5 ε}{4} . \end{aligned} \end{array}$ Finally setting $ψ_{ε} = ϕ_{4 ε / 3}$ , we deduce from (3.11)–(3.13) that (3.1)–(3.4) are satisfied with $\begin{array}{l} c_{0} = \frac{16}{9} t_{0}, K_{1} = {(\frac{4}{3})}^{α} ‖ v_{0} ‖_{\infty}, \end{array}$ and assertion (i) is proved with $ε_{0} = \frac{4}{3} d$ .

(ii) The argument is similar, now considering the problem $\begin{array}{l} (3.14) & \{\begin{matrix} v_{t} - v_{x x} = | v_{x} |^{p}, & x \in (0, 2), t > 0, \\ v = 0, & x \in {0, 1}, t > 0, \\ v (x, 0) = K ψ (x), & x \in (0, 2), \end{matrix} \end{array}$ where $K > 0$ and $ψ \in C_{0}^{\infty} (R)$ is symmetric with respect to $x = 1$ , nontrivial with $Supp (ψ) \subset (1 - η, 1 + η)$ and $0 ⩽ ψ ⩽ 1$ . Denote by v the unique global viscosity solution of (3.9) and by $T_{0} : = T (v_{0})$ its maximal existence time as a classical solution. By [33, Theorem 4], we may take $K = \tilde{K} (η) > 0$ sufficiently large, such that $T_{0} < \infty$ and that there exist $c_{2} > c_{1} > T_{0}$ and $C_{0} > 0$ such that $v (0, s) > C_{0}$ for all $s \in [c_{1}, c_{2}]$ . Now consider $\begin{array}{l} v_{ε} (x, t) = ε^{α} v (ε^{- 1} x, ε^{- 2} t), (x, t) \in [0, 2 ε] \times [0, \infty) . \end{array}$ Then $v_{ε}$ solves problem (1.1) with $(0, 2)$ replaced by $(0, 2 ε)$ and initial data $v_{ε} (x, 0) : = K ε^{α} ψ (ε^{- 1} x)$ . Under assumption (3.5), since $Supp (v_{ε} (\cdot, 0)) \subset ((1 - η) ε, (1 + η) ε)$ and $v_{ε} (\cdot, 0) ⩽ K ε^{α}$ , we have $ϕ ⩾ v_{ε} (\cdot, 0)$ in $(0, 2 ε)$ . By Proposition 2.2, using $u ⩾ 0$ , it follows that $u ⩾ v_{ε}$ in $[0, 2 ε] \times (0, \infty)$ , hence $u (0, s ε^{2}) ⩾ v_{ε} (0, s ε^{2}) = ε^{α} v (0, s) = C_{0} ε^{α}$ for all $s \in [c_{1}, c_{2}]$ . □

For the proof of Proposition 3.2, we need two lemmas. Our first simple lemma gives a pointwise control from above with linear growth in time.
Lemma 3.3.
Let $h \in X \cap C^{2} (\overline{Ω})$ and set $\begin{array}{l} (3.15) & λ = λ_{h} : = sup_{\overline{Ω}} [Δ h + | \nabla h |^{p}] . \end{array}$ For any $γ ⩾ 0$ , if $ϕ \in X$ satisfies $ϕ ⩽ h + γ$ in Ω, then $\begin{array}{l} u ⩽ h + γ + λ t in Ω \times (0, \infty) . \end{array}$
Proof.
Set $\overline{u} (x, t) = h (x) + γ + λ t$ . We have $\begin{array}{l} P \overline{u} : = {\overline{u}}_{t} - Δ \overline{u} - | \nabla \overline{u} |^{p} = λ - Δ h - | \nabla h |^{p} ⩾ 0 in \overline{Ω} \times (0, \infty) . \end{array}$ Since also $\overline{u} (0, \cdot) ⩾ ϕ$ by assumption and $\overline{u} ⩾ 0$ on $\partial Ω \times (0, \infty)$ , the conclusion follows from Proposition 2.2. □

Our second lemma shows that the solution becomes classical again after some short time, provided it is suitably controlled in amplitude2
²
We stress that, unlike in more standard criteria, we do not assume any control of the gradient of the solution near the boundary.

near the boundary. It is a localized version (in space near the boundary and in time) of an observation from [35]. Lemma 3.4.
There exist $K_{2}, K_{3} > 0$ depending only on $Ω, p$ such that if, for some $ε, η, s > 0$ , $\begin{array}{l} (3.16) & u (\cdot, s) ⩽ η in Ω_{ε}, \end{array}$ and (in case $Γ_{ε} \neq \emptyset$ ) $\begin{array}{l} (3.17) & u ⩽ K_{2} ε in Γ_{ε} \times [s, \hat{s}], where \hat{s} = s + K_{3} η, \end{array}$ then $\begin{array}{l} u is a classical solution for t > \hat{s} close to \hat{s} . \end{array}$
Proof.
We modify a comparison argument from [35] (see [35, Lemma 3.2]). Let $ψ \in C^{2} (\overline{Ω})$ be the unique solution of the linear problem (2.9) and let $c_{1} > 0$ be the constant of inequality (2.10). Let $κ = {(2 ‖ \nabla ψ ‖_{\infty})}^{- p / (p - 1)}$ , $K_{3} = κ^{- 1}$ , $\hat{s} = s + K_{3} η$ , and define $\begin{array}{l} w (x, t) = η + κ [2 ψ (x) + s - t], (x, t) \in \overline{Q}, with Q : = Ω_{ε} \times (s, \hat{s}] . \end{array}$ We compute $\begin{array}{l} w_{t} - Δ w - | \nabla w |^{p} = - κ + 2 κ - {(2 κ | \nabla ψ |)}^{p} ⩾ 0 in Q . \end{array}$ On the other hand, by the choice of $\hat{s}$ , we have $w ⩾ 2 κ ψ$ in Q, hence in particular $w ⩾ 0$ on $\partial Ω \times (s, \hat{s})$ and, using (2.10), $w ⩾ 2 κ c_{1} ε$ on $Γ_{ε} \times (s, \hat{s})$ . Moreover we have $w (x, s) ⩾ η$ in $Ω_{ε}$ . Choosing $K_{2} = 2 κ c_{1}$ , in view of assumptions (3.16)–(3.17), it follows from Proposition 2.2 that $u ⩽ w$ in Q. In particular, we have $u (\cdot, \hat{s}) ⩽ 2 κ ψ$ in $Ω_{ε}$ . Since also $u ⩽ ‖ ϕ ‖_{\infty}$ in $\overline{Ω} \times [0, \infty)$ by Proposition 2.2, using (2.10) we get $\begin{array}{l} u (\cdot, \hat{s}) ⩽ C_{1} ψ in Ω, with C_{1} : = max (2 κ, {(c_{1} ε)}^{- 1} ‖ ϕ ‖_{\infty}) . \end{array}$ It then follows from Proposition 2.3 and uniqueness of the viscosity solution that u is a classical solution for $t > \hat{s}$ close to $\hat{s}$ . □
Proof of Proposition 3.2.
Fix $θ_{ε} \in C^{2} (\overline{Ω})$ such that $θ_{ε} = 0$ in ${\overline{Ω}}_{ε / 2}$ , $θ_{ε} = 1$ in ${\overline{Ω}}^{ε}$ and $0 ⩽ θ_{ε} ⩽ 1$ . Setting $h = θ_{ε} H$ , assumption (3.8) guarantees that $\begin{array}{l} ϕ ⩽ γ + h in Ω . \end{array}$ It follows from Lemma 3.3 that $\begin{array}{l} u (x, t) ⩽ γ + h (x) + λ t in Ω \times (0, \infty), \end{array}$ where $λ = λ_{h}$ is given by (3.15), hence in particular $\begin{array}{l} (3.18) & u (x, t) ⩽ γ + λ t in {\overline{Ω}}_{ε / 2} \times (0, \infty) . \end{array}$ Let $\begin{array}{l} (3.19) & \overline{γ} : = \frac{1}{4} min {1, {(λ K_{3})}^{- 1}} K_{2} ε, \end{array}$ where $K_{2}, K_{3}$ are given by Lemma 3.4, and note that $\overline{γ}$ is uniformly positive for ε fixed and $‖ H ‖_{C^{2} ({\overline{Ω}}^{ε / 2})}$ bounded. Assume $\begin{array}{l} (3.20) & 0 < γ ⩽ \overline{γ} \end{array}$ and take any τ such that $\begin{array}{l} (3.21) & K_{3} γ ⩽ τ ⩽ K_{3} \overline{γ} . \end{array}$ Set $s = {(1 + K_{3} λ)}^{- 1} (τ - K_{3} γ) \in [0, τ)$ and $η = γ + λ s$ , hence $\begin{array}{l} (3.22) & s + K_{3} η = τ . \end{array}$ By (3.18) we have $\begin{array}{l} (3.23) & u (\cdot, s) ⩽ η in Ω_{ε / 2} . \end{array}$ Moreover, since (3.19)–(3.21) guarantee that $τ ⩽ \frac{1}{4} λ^{- 1} K_{2} ε ⩽ λ^{- 1} (\frac{1}{2} K_{2} ε - γ)$ , hence $γ + λ τ ⩽ K_{2} ε / 2$ , inequality (3.18) also implies $\begin{array}{l} (3.24) & u (x, t) ⩽ γ + λ τ ⩽ K_{2} ε / 2 in Γ_{ε / 2} \times [s, τ] . \end{array}$ It then follows from (3.22)–(3.24) and Lemma 3.4 that u is a classical solution for $t > τ$ close to τ. Since this is true for any τ satisfying (3.21), we have shown that u is classical on $(K_{3} γ, K_{3} \overline{γ}]$ .

Finally set $γ_{0} = \overline{γ} / 2$ and $L = 2 K_{3}$ . If ϕ satisfies (3.8) with $γ \in [0, γ_{0})$ , then the above implies that u is classical on $(K_{3} γ, K_{3} \overline{γ}] = (L γ / 2, L γ_{0}] \supset [L γ, L γ_{0}]$ . The proposition is proved. □

4. Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1.
Let $m ⩾ 1$ . We shall construct a suitable, multibump initial data of the form $ϕ = \sum_{i = 1}^{m} ϕ_{i}$ , with compactly supported $ϕ_{i}$ . Recall that the constants $ε_{0}, K_{1}, L, c_{0}$ are defined in Propositions 3.1 and 3.2. In view of the application of Proposition 3.2, bumps will be sequentially added by moving toward the boundary. Therefore, we proceed by backward induction and recursively define functions $ϕ_{1}, \dots, ϕ_{m} \in C^{2} (\overline{Ω})$ and numbers $ε_{1} > \dots > ε_{m} > 0$ as follows.

Fix $ε_{m} \in (0, ε_{0})$ and set $\begin{array}{l} (4.1) & ϕ_{m} = ψ_{ε_{m}}, \end{array}$ where $ψ_{ε_{m}}$ is given by Proposition 3.1(i). Take $i \in {2, \dots, m}$ and assume that $ϕ_{i}, \dots, ϕ_{m} \in C^{2} (\overline{Ω})$ and $ε_{m} > \dots > ε_{i} > 0$ have already been chosen. Set $\begin{array}{l} h_{i} : = \sum_{j = i}^{m} ϕ_{j} \in C^{2} (\overline{Ω}) \end{array}$ and $\begin{array}{l} (4.2) & γ_{i} = min {\frac{1}{2} γ_{0} (Ω, p, ε_{i}, ‖ h_{i} ‖_{C^{2} (\overline{Ω})}), \frac{1}{4} c_{0} L^{- 1} ε_{i}^{2}}, \end{array}$ where the function $γ_{0}$ is given by Proposition 3.2. We next choose $\begin{array}{l} (4.3) & ε_{i - 1} = min {{(K_{1}^{- 1} γ_{i})}^{1 / α}, \frac{1}{2} {(c_{0}^{- 1} L γ_{i})}^{1 / 2}} < \frac{1}{2} ε_{i} \end{array}$ and we set $ϕ_{i - 1} = ψ_{ε_{i - 1}}$ where $ψ_{ε_{i - 1}}$ is given by Proposition 3.1(i). Recalling (4.1), we thus have $\begin{array}{l} (4.4) & ϕ_{i} = ψ_{ε_{i}}, i = 1, \dots, m \end{array}$ and $\begin{array}{l} (4.5) & ε_{1} > \dots > ε_{m} > 0 . \end{array}$

Now set $\begin{array}{l} (4.6) & ϕ : = \sum_{j = 1}^{m} ϕ_{j} \end{array}$ and $\begin{array}{l} s_{i} : = c_{0} ε_{i}^{2}, i = 1, \dots, m, \\ {\hat{s}}_{i} : = L γ_{i} \in [4 s_{i - 1}, \frac{1}{4} s_{i}], i = 2, \dots, m \end{array}$ (where we used (4.2), (4.3)). Let us verify that the initial data ϕ has all the desired properties. Since $ϕ ⩾ ϕ_{i}$ for each $i \in {1, \dots, m}$ , it follows from (3.4), (4.4) and Proposition 2.2 that $\begin{array}{l} (4.7) & sup_{x \in \partial Ω} u (ϕ; \cdot, s_{i}) ⩾ sup_{x \in \partial Ω} u (ϕ_{i}; \cdot, s_{i}) > 0, i \in {1, \dots, m} . \end{array}$ Next for each $i \in {2, \dots, m}$ , by (3.1), (3.2), (4.3)–(4.5), we have $\begin{array}{l} ϕ = \underset{0 in Ω^{ε_{i}}}{\underset{︸}{\sum_{j = 1}^{i - 1} ϕ_{j}}} + \sum_{j = i}^{m} ϕ_{j} = h_{i} in Ω^{ε_{i}} \end{array}$ and $\begin{array}{l} ϕ ⩽ K_{1} ε_{i - 1}^{α} + \underset{0 in Ω_{ε_{i}}}{\underset{︸}{\sum_{j = i}^{m} ϕ_{j}}} = K_{1} ε_{i - 1}^{α} ⩽ γ_{i} in Ω_{ε_{i}} . \end{array}$ This along with (4.2) and Proposition 3.2 guarantees that $\begin{array}{l} (4.8) & u is a classical solution for t \in (L γ_{i}, 2 L γ_{i}] = ({\hat{s}}_{i}, 2 {\hat{s}}_{i}], i \in {2, \dots, m} . \end{array}$ In view of (4.7)–(4.8) and $s_{i - 1} < {\hat{s}}_{i} < 2 {\hat{s}}_{i} < s_{i}$ , the theorem is thus proved.

For future reference, we note that since $ϕ = 0$ in $Ω_{ε_{1}}$ , (4.2) and Proposition 3.2 also guarantee that $\begin{array}{l} (4.9) & u is a classical solution for t \in (0, 2 {\hat{s}}_{1}) . \end{array}$ □

We next turn to the proof of Theorem 1.2. We shall use the following uniform lower bound from [17] on the boundary gradient blow-up profile of maximal classical solutions. First recall that, thanks to the regularity of Ω, we can find $δ_{0} > 0$ such that every $x \in Ω_{δ_{0}}$ has a unique projection $P (x)$ onto $\partial Ω$ . In this way, we can in particular extend the outer normal vector field to $Ω_{δ_{0}}$ by setting $ν_{x} = ν_{P (x)}$ , and we then denote $u_{ν} (x, t) = \nabla u (x, t) \cdot ν_{x}$ for all $(x, t) \in Ω_{δ_{0}} \times (0, \infty)$ .
Proposition 4.1.
Let $p > 2$ , $ε \in (0, 1)$ and $M > 0$ . There exists $η = η (p, Ω, M, ε) \in (0, δ_{0})$ such that for any $ϕ \in X \cap C^{2} (\overline{Ω})$ with $‖ ϕ ‖_{C^{2} (\overline{Ω})} ⩽ M$ and $T : = T (ϕ) < \infty$ , and any GBU point a of u (i.e., ${lim sup}_{t \to T^{-}, x \to a} | \nabla u (x, t) | = \infty$ ), then $\begin{array}{l} (4.10) & r^{1 / (p - 1)} u_{ν} (a + r ν_{a}, T) ⩽ - (1 - ε) d_{p} for all r \in (0, η] . \end{array}$

Proposition 4.1 was essentially proved in [17] (see Proposition 5.2 and estimate (1.13)), except for the uniform dependence of η. The latter can be easily checked along the proof, using the fact that the Bernstein estimate $\begin{array}{l} (4.11) & | \nabla u (x, t) | ⩽ C_{1} δ^{- 1 / (p - 1)} (x) + C_{2} in Ω \times [0, T) \end{array}$ from [40, Theorem 3.2] and the time derivative estimate ${sup}_{Ω \times (0, \infty)} | u_{t} | ⩽ C_{3}$ (see [40, Proposition 2.4]), hold with uniform constants $C_{1} = C_{1} (n, p)$ , $C_{2} = C_{2} (p, Ω, M)$ and $C_{3} = C_{3} (p, Ω, M)$ . Proof of Theorem 1.2.
It is based on a modification of the proof of Theorem 1.1 and a limiting argument. Let $m ⩾ 2$ and let $ϕ_{i}$ , $s_{i}$ and ${\hat{s}}_{i}$ be as the proof of Theorem 1.1, which satisfy $s_{i - 1} < {\hat{s}}_{i} < 2 {\hat{s}}_{i} < s_{i}$ for all $i \in {2, \dots, m}$ . We define the family of initial data $\begin{array}{l} (4.12) & Φ_{λ} : = λ ϕ_{1} + \sum_{j = 2}^{m} ϕ_{j}, λ \in [0, 1], \end{array}$ and denote by $u_{λ}$ the corresponding global viscosity solution of (1.1). For any $λ \in [0, 1]$ , by the arguments leading to (4.7) and (4.8), we have $\begin{array}{l} (4.13) & sup_{x \in \partial Ω} u_{λ} (\cdot, s_{i}) > 0, i \in {2, \dots, m} \end{array}$ and $\begin{array}{l} (4.14) & u_{λ} is a classical solution for t \in ({\hat{s}}_{i}, 2 {\hat{s}}_{i}) and i \in {2, \dots, m} . \end{array}$ Also, for $λ = 1$ , we have $\begin{array}{l} (4.15) & sup_{x \in \partial Ω} u_{1} (\cdot, s_{1}) > 0 \end{array}$ whereas, for $λ = 0$ , since $Φ_{0} = \sum_{j = 2}^{m} ϕ_{j}$ , we may apply (4.9) with $ε_{1}, {\hat{s}}_{1}$ replaced by $ε_{2}, {\hat{s}}_{2}$ to get that $\begin{array}{l} (4.16) & u_{0} is a classical solution for t \in (0, 2 {\hat{s}}_{2}) . \end{array}$

Next let $\begin{array}{l} λ^{} = inf E, E : = {λ \in [0, 1]; T (Φ_{λ}) < {\hat{s}}_{2}} . \end{array}$ Note that $1 \in E$ by (4.15), so that $λ^{} \in [0, 1]$ is well defined. If $λ^{} > 0$ then, for all $λ \in [0, λ^{})$ , we have $u_{λ} = 0$ on $\partial Ω \times [0, {\hat{s}}_{2}]$ , hence $\begin{array}{l} u_{λ} = 0 on \partial Ω \times [0, 2 {\hat{s}}_{2}] \end{array}$ by (4.14). By continuous dependence (cf. (2.3)), it follows that $\begin{array}{l} (4.17) & u_{λ^{}} = 0 on \partial Ω \times [0, 2 {\hat{s}}_{2}] . \end{array}$ Moreover, (4.17) remains true in case $λ^{} = 0$ due to (4.16).

On the other hand, by definition, there exists a sequence $λ_{j} \to λ^{}$ with $λ_{j} ⩾ λ^{}$ , such that $T_{j} : = T (Φ_{λ_{j}}) < {\hat{s}}_{2}$ . Since $\partial Ω$ is compact, $u_{λ_{j}}$ admits at least a GBU point $a_{j} \in \partial Ω$ . Observing that ${sup}_{λ \in [0, 1]} ‖ Φ_{λ} ‖_{C^{2} (\overline{Ω})} < \infty$ , we may apply (4.10) with $ε = \frac{1}{2}$ and, after integrating along the normal direction, we obtain $\begin{array}{l} u_{λ_{j}} (a_{j} + r ν_{a_{j}}, T_{j}) ⩾ \frac{c_{p}}{2} r^{α} for all r \in (0, η] . \end{array}$ After extracting a subsequence, we may assume that $a_{j} \to a \in \partial Ω$ and $T_{j} \to t_{0} \in [0, {\hat{s}}_{2}]$ . Passing to the limit by means of (2.3) and of the continuity of $u_{λ^{}}$ , we deduce that $\begin{array}{l} u_{λ^{}} (a + r ν_{a}, t_{0}) ⩾ \frac{c_{p}}{2} r^{α} for all r \in (0, η], \end{array}$ hence in particular $u_{λ^{}} (\cdot, t_{0}) \notin C^{1} (\overline{Ω})$ . Therefore $T (Φ_{λ^{}}) ⩽ t_{0} ⩽ {\hat{s}}_{2}$ . Taking (4.17) into account, it follows that $u_{λ^{}}$ undergoes gradient blow-up at $T (Φ_{λ^{}})$ without loss of boundary conditions. In view of (4.13), (4.14), this completes the proof (since $m ⩾ 2$ is arbitrary). □

5. Estimates for $n = 1$

Let us first recall the following estimates from [34] (see [34, Lemmas 5.1–5.4]).

Lemma 5.1.
Let $ϕ \in X_{1}$ and $t_{0} > 0$ . There exists a constant $K > 0$ (depending on ϕ and $t_{0}$ ) with the following properties.

(i) For all $t ⩾ t_{0}$ , $\begin{array}{l} (5.1) & u_{x x} (x, t) ⩽ K, 0 < x < 1, \\ (5.2) & u_{x} (x, t) ⩽ U_{x}^{} (x) + K x, 0 < x < 1, \\ (5.3) & u_{x} (x, t) ⩾ - U_{x}^{} (1 - x) - K (1 - x), 0 < x < 1 . \end{array}$ In particular, there exists $\bar{x} \in (0, \frac{1}{2})$ (depending on ϕ and $t_{0}$ ) such that $\begin{array}{l} (5.4) & u_{x} (x, t) ⩾ - U_{x}^{} (x), 0 < x < \bar{x} . \end{array}$

(ii) Let* $t ⩾ t_{0}$ and assume that there exists a sequence $(x_{j}, t_{j}) \to (0, t)$ such that $u_{x} (x_{j}, t_{j}) \to \infty$ . Then $\begin{array}{l} (5.5) & | u (x, t) - u (0, t) - U^{} (x) | ⩽ K x^{2}, 0 < x ⩽ \frac{1}{2}, \\ (5.6) & | u_{x} (x, t) - U_{x}^{} (x) | ⩽ K x, 0 < x ⩽ \frac{1}{2} . \end{array}$ Moreover, ( 5.5 )–( 5.6 ) are true whenever $u (0, t) > 0$ .

We shall use also the following simple lemma, which connects the sign of $u_{t}$ near the boundary with the sign of $u_{x} - U_{x}^{}$ and is a variant of [34, Lemma 5.6].
Lemma 5.2.
Let* $ϕ \in X_{1}$ , $t > 0$ and $a \in (0, \frac{1}{2})$ . Then $\begin{array}{l} (5.7) & u_{t} (\cdot, t) ⩽ 0 in (0, a) ⟹ u_{x} (\cdot, t) ⩽ U_{x}^{} in (0, a) \end{array}$ and* $\begin{array}{l} (5.8) & u_{t} (\cdot, t) ⩾ 0 in (0, a) and u_{x} (0, t) = \infty ⟹ u_{x} (\cdot, t) ⩾ U_{x}^{} in (0, a) . \end{array}$
Proof.
Set $v (x) = u (x, t)$ .

First assume $u_{t} (\cdot, t) ⩽ 0$ in $(0, a)$ . Thus we have $- v_{x x} ⩾ | v_{x} |^{p}$ in $(0, a)$ . In particular $v_{x}$ is nonincreasing, and we may assume ${lim}_{x \to 0} v_{x} > 0$ since otherwise the conclusion is obvious. Therefore the set ${x \in (0, a); v_{x} > 0}$ is an interval of the form $(0, b)$ with $b \in (0, a]$ . Then we have $\begin{array}{l} {[{(v_{x})}^{1 - p}]}_{x} = - (p - 1) {(v_{x})}^{- p} v_{x x} ⩾ p - 1, 0 < x < b . \end{array}$ By integration, it follows that $\begin{array}{l} {(v_{x})}^{1 - p} (x) ⩾ {(v_{x})}^{1 - p} (y) + (p - 1) (x - y) ⩾ (p - 1) (x - y), 0 < y < x < b . \end{array}$ Letting $y \to 0$ , we obtain $\begin{array}{l} v_{x} (x) ⩽ {[(p - 1) x]}^{- 1 / (p - 1)} = U_{x}^{} (x), 0 < x < b, \end{array}$ which gives the desired conclusion.

Next assume $u_{t} (\cdot, t) ⩾ 0$ in $(0, a)$ and $u_{x} (0, t) = \infty$ . By (1.5) we have $\begin{array}{l} (5.9) & lim_{y \to 0} u_{x} (y, t) = \infty, \end{array}$ hence in particular $v_{x} > 0$ for $x > 0$ small. Let $\begin{array}{l} b : = sup {x_{0} \in (0, a]; v_{x} > 0 on (0, x_{0})} \in (0, a] . \end{array}$ Since $- v_{x x} ⩽ {(v_{x})}^{p}$ in $(0, b)$ , the same calculation as above now yields $\begin{array}{l} {(v_{x})}^{1 - p} (x) ⩽ {(v_{x})}^{1 - p} (y) + (p - 1) (x - y), 0 < y < x < b . \end{array}$ Letting $y \to 0$ and using (5.9), we get $v_{x} ⩾ U_{x}^{}$ in $(0, b)$ . In view of the definition of b, this in turn implies $b = a$ . The lemma is proved. □

The following lemma will enable us to compare $u_{x}$ and $U_{x}^{}$ in the singular regime.
Lemma 5.3.
Let $ϕ \in X_{1}$ , $t_{2} > t_{1} > 0$ and $a \in (0, \frac{1}{2})$ . Set $\begin{array}{l} Q : = (0, a) \times (t_{1}, t_{2}], Γ : = ((0, a] \times {t_{1}}) \cup ({a} \times (t_{1}, t_{2}]) . \end{array}$

(i) Assume that $\begin{array}{l} (5.10) & u_{x} - U_{x}^{} ⩽ 0 on Γ . \end{array}$ Then* $\begin{array}{l} (5.11) & u_{x} - U_{x}^{} ⩽ 0 in Q . \end{array}$

(ii) Assume that* $\begin{array}{l} (5.12) & u_{x} - U_{x}^{} ⩾ 0 on Γ \end{array}$ and* $\begin{array}{l} (5.13) & u_{x} (0, s) = \infty for all s \in [t_{1}, t_{2}] . \end{array}$ Then $\begin{array}{l} (5.14) & u_{x} - U_{x}^{} ⩾ 0 in Q . \end{array}$
Proof.
First observe that if the function $W : = u_{x} - U_{x}^{}$ attains a local extremum at some point $(x, t) \in Q$ , then at this point we have $u_{x x} = U_{x x}^{}$ hence $\begin{array}{l} W_{t} - W_{x x} = h (u_{x}) u_{x x} - h (U_{x}^{}) U_{x x}^{} = [h (u_{x}) - h (U_{x}^{})] U_{x x}^{} \end{array}$ where $h (s) = p | s |^{p - 2} s$ , that is, $\begin{array}{l} (5.15) & W_{t} - W_{x x} = - [h (W + U_{x}^{}) - h (U_{x}^{})] {(U_{x}^{})}^{p} . \end{array}$

Let us first assume (5.10). Suppose for contradiction that (5.11) fails, i.e. ${sup}_{Q} W \in (0, \infty]$ . By (5.2) in Lemma 5.1, we have $W ⩽ K x$ in Q. Therefore, W must attain a positive maximum at some point $(x, t) \in Q$ . Since h is an increasing function, at this point we have $h (W + U_{x}^{}) > h (U_{x}^{})$ , hence (5.15) yields $0 ⩽ W_{t} - W_{x x} < 0$ : a contradiction.

Let us next assume (5.12) and (5.13). Suppose for contradiction that (5.14) fails, i.e. ${inf}_{Q} W \in [- \infty, 0)$ . In view of (5.13), estimate (5.6) in Lemma 5.1 guarantees that $W ⩾ - K x$ in Q. Therefore, W must attain a negative minimum at some point $(x, t) \in Q$ and we reach a contradiction similarly as before.

The lemma is proved. □

The next lemma shows that the solutions $u_{k}$ of the truncated problems (2.1) have good enough approximation properties in the singular regime too. Namely $u_{k}$ has large space derivative near 0 whenever $u_{x} (0, t) = \infty$ and k is large. Lemma 5.4.
Let $ϕ \in X_{1}$ . Then we have $\begin{array}{l} lim_{\begin{array}{c} k \to \infty \\ η \to 0 \end{array}} [inf {\partial_{x} u_{k} (x, t); x \in [0, η], t \in S}] = \infty, \end{array}$ where $S$ is defined in ( 1.7 ).
Proof.
Let $M, k_{0}$ be given by (2.5) with $t_{0} = T$ . By (2.2), there exist $A_{0} > 0$ and $k_{1} ⩾ k_{0}$ such that $F_{k} (s) > M$ whenever $s ⩾ A_{0}$ and $k ⩾ k_{1}$ . Let $A ⩾ A_{0}$ . Then it follows from (2.1), (2.5) that, for any $k ⩾ k_{1}$ and $(x, t) \in [0, 1] \times [T, \infty)$ , $\begin{array}{l} (5.16) & \partial_{x} u_{k} (x, t) ⩾ A ⟹ [\partial_{x}^{2} u_{k}] (x, t) < 0 ⟹ inf_{[0, x]} \partial_{x} u_{k} (\cdot, t) ⩾ A . \end{array}$ On the other hand, by (5.6), for any $t \in S$ , we have $\begin{array}{l} u_{x} (x, t) ⩾ U_{x}^{*} (x) - K x, 0 < x < \frac{1}{2} . \end{array}$ Therefore there exists $η > 0$ such that, for any $t \in S$ , $u_{x} (η, t) ⩾ A + 1$ . Moreover, we know from [35] that there exists $\tilde{T} \in (T, \infty)$ such that $S \subset [T, \tilde{T}]$ . Since $\partial_{x} u_{k} \to u_{x}$ locally uniformly in $(0, 1) \times [T, \tilde{T}]$ , there exists $k_{2} ⩾ k_{1}$ such that for all $k ⩾ k_{2}$ , $\begin{array}{l} \partial_{x} u_{k} (η, t) ⩾ A for any t \in S, \end{array}$ hence ${inf}_{[0, η] \times S} \partial_{x} u_{k} ⩾ A$ by (5.16). This proves the lemma. □

6. Zero number properties I: Monotonicity

We start with the following intersection properties. Note that these properties are well known for classical solutions, but are not standard in the context of viscosity solutions.

Proposition 6.1.
Let $ϕ \in X_{1}$ .

(i) For any $t_{0} \in (0, T (ϕ))$ , we have $\begin{array}{l} (6.1) & 0 ⩽ N (t) ⩽ N (t_{0}) < \infty, t > t_{0} . \end{array}$ Moreover, if $N (0) < \infty$ then we may take $t_{0} = 0$ in ( 6.1 ).

(ii) For any $τ_{2} > τ_{1} > 0$ such that $u \neq c_{p}$ on ${0} \times [τ_{1}, τ_{2}]$ , the function N is nonincreasing and right continuous on $[τ_{1}, τ_{2}]$ . Moreover, if $‖ ϕ ‖_{\infty} ⩽ c_{p}$ , then N is nonincreasing and right continuous on $[0, \infty)$ .

(iii) For all $t \in T$ with $u (1, t) \neq c_{p}$ , the limits $N (t^{\pm}) : = {lim}_{s \to t^{\pm}} N (s)$ are well defined and finite and we have $\begin{array}{l} (6.2) & N (t^{-}) ⩾ N (t) = N (t^{+}) . \end{array}$

We shall also establish a monotonicity property for the number of intersections of two solutions, which will be used in the proof of Theorem 1.4. Actually, we will need to compare solutions of problem (1.1) on different space intervals. To this end, for any $b \in (0, 1]$ , we set $\begin{array}{l} X_{b} : = {ϕ \in C^{1} ([0, b]); ϕ ⩾ 0, ϕ (0) = ϕ (b) = 0} \end{array}$ and the number of sign changes $z_{| [0, b]}$ is defined similarly as in (1.4), replacing 1 with b. Also in this and the next section, beside the singular steady state $U^{}$ , we shall also use the regular steady states $U_{a}$ , such that $U_{a} (0) = 0$ and $U_{a}^{'} (0) = a$ , with $a > 0$ . Namely, $\begin{array}{l} (6.3) & U_{a} (x) = U^{} (x + k) - U^{} (k), where k = a^{1 - p} / (p - 1) . \end{array}$
Proposition 6.2.
Let* $b \in (0, 1]$ , $ϕ_{1} \in X_{1}$ , $ϕ_{2} \in X_{b}$ . Denote by $u, v$ the corresponding viscosity solutions of ( 1.1 ), respectively with $Ω = (0, 1)$ and $Ω = (0, b)$ . Assume that, for all $t ⩾ 0$ , $\begin{array}{l} (6.4) & \{\begin{matrix} u (\cdot, t) ≢ v (\cdot, t) on [0, 1], & if b = 1, \\ v (b, t) = 0 (in the classical sense), & if b \in (0, 1), \end{matrix} \end{array}$ and set $\begin{array}{l} N (t) = z_{| [0, b]} (u (\cdot, t) - v (\cdot, t)), t ⩾ 0 . \end{array}$ Then $\begin{array}{l} (6.5) & N (t) is finite and nonincreasing on (0, \infty) . \end{array}$ If moreover $N (ϕ_{1} - ϕ_{2}) < \infty$ , then ( 6.5 ) remains true on $[0, \infty)$ .

The next result shows that the assumption $‖ ϕ ‖_{\infty} ⩽ c_{p}$ in Proposition 6.1 is not purely technical. We actually suspect that for any $M > c_{p}$ , the monotonicity of $N (t)$ fails for some $ϕ \in X_{1}$ with $‖ ϕ ‖_{\infty} = M$ .
Proposition 6.3.
There exists $ϕ_{1} \in X_{1}$ , with $‖ ϕ ‖_{\infty} > c_{p}$ and $T = T (ϕ) < \infty$ , and there exist $t_{2} > t_{1} > T$ such that $N (t_{1}) = 0$ and $N (t_{2}) = 1$ .

In the proof of Proposition 6.1 we shall use the following two lemmas.
Lemma 6.4.
Let $a, b \in R$ , $a < b$ and assume that $v \in C (a, b)$ , with $v ≢ 0$ , is such that $z (v) < \infty$ . Let ${(v_{j})}_{j}$ be a sequence of $C (a, b)$ such that $v_{j} \to v$ in $C_{loc} (a, b)$ . Then ${lim inf}_{j \to \infty} z (v_{j}) ⩾ z (v)$ .
Proof.
Let $m = z (v)$ . By assumption there exist $a < x_{0} < \dots < x_{m} < b$ such that $v (x_{i - 1}) v (x_{i}) < 0$ for $i = 1, \dots, m$ . Since $v_{j} \to v$ in $C ([x_{0}, x_{m}])$ , there exists $j_{0}$ such that, for all $j ⩾ j_{0}$ , we have $v_{j} (x_{i - 1}) v_{j} (x_{i}) < 0$ for $i = 1, \dots, m$ , hence $z (v_{j}) ⩾ m$ . The lemma follows. □
Lemma 6.5.
Let $ϕ \in X_{1}$ and let $0 < t_{0} < t_{1} < T : = T (ϕ)$ . Then $N (t_{0}) < \infty$ and there exists $a_{1} > 0$ such that $\begin{array}{l} (6.6) & 0 ⩽ z (u (\cdot, t_{1}) - U_{a}) ⩽ N (t_{0}), a ⩾ a_{1} . \end{array}$ Moreover ( 6.6 ) remains true with $t_{0} = 0$ provided $N (0) < \infty$ .
Proof.
Set $U_{\infty} : = U^{}$ . Let us first observe that, for any $a \in (0, \infty]$ , we have $\begin{array}{l} (6.7) & u (\cdot, t) - U_{a} ≢ 0 on [0, 1], for each t ⩾ 0 \end{array}$ (i.e., $0 ⩽ z (u (\cdot, t) - U_{a}) ⩽ \infty$ ). Indeed, this is obvious if $u (1, t) \neq U_{a} (1)$ whereas, if $u (1, t) = U_{a} (1) > 0$ then, by applying Lemma 5.1(ii) to the solution $w (x, t) = u (1 - x, t)$ , we get ${lim}_{y \to 1} u_{x} (y, t) = - \infty$ , hence $u (y, t) \neq U_{a} (y)$ for $y < 1$ close to 1.

Fix $t_{0} \in (0, T)$ , or $t_{0} \in [0, T)$ if $N (0) < \infty$ , and let $t_{1} \in (t_{0}, T)$ . Since $\begin{array}{l} sup_{s \in [0, t_{1}]} {‖ u_{x} (s) ‖}_{\infty} < \infty, \end{array}$ there exist $η \in (0, 1)$ and $a_{0} > 0$ such that $\begin{array}{l} (6.8) & u < U_{a_{0}} < U^{} in (0, η] \times [0, t_{1}] . \end{array}$ Therefore, we have $N (s) = z_{| [η, 1]} (u (\cdot, s) - U^{})$ for all $s \in [0, t_{1}]$ . Since $U^{}$ is smooth on $[η, 1]$ and $U^{} (1) > 0$ , it follows from standard properties of the zero number (cf. [3] and see also [34, Proposition 6.1]) that N is finite and nonincreasing on $[t_{0}, t_{1}]$ and there exists $t_{2} \in (0, t_{1})$ such that all zeros of $u (\cdot, t_{2}) - U^{}$ in $[η, 1]$ are nondegenerate. Since $U_{a}$ increases with a and $U_{a} \to U^{}$ in $C^{1} ([η, 1])$ as $a \to \infty$ , recalling (6.8), there exists $a_{1} > a_{0}$ such that $\begin{array}{l} (6.9) & z (u (\cdot, t_{2}) - U_{a}) = z (u (\cdot, t_{2}) - U^{}) = N (t_{2}) ⩽ N (t_{0}), a ⩾ a_{1} . \end{array}$ On the other hand, since u and $U_{a}$ are smooth on $[0, 1] \times (0, T)$ with $u = U_{a} = 0$ at $x = 0$ and $u = 0 < U_{a}$ at $x = 1$ , the zero number principle guarantees that $z (u (\cdot, t_{2}) - U_{a})$ is finite and nonincreasing on $(t_{0}, t_{1}]$ . This along with (6.9) yields the desired conclusion. □
Proof of Proposition 6.1(i).
First of all, we have $0 ⩽ N (t) ⩽ \infty$ by (6.7). Set $T = T (ϕ)$ and fix $t_{0} \in (0, T)$ , or $t_{0} \in [0, T)$ in case $N (0) < \infty$ . Let $t > t_{0}$ and pick any $t_{1} \in (t_{0}, min (T, t))$ . By Lemma 6.5, we have $\begin{array}{l} (6.10) & z (u (\cdot, t_{1}) - U_{a}) ⩽ N (t_{0}) < \infty, a ⩾ a_{1} . \end{array}$ Next, since ${sup}_{s \in [0, t_{1}]} ‖ u_{x} (s) ‖_{\infty} < \infty$ , there exists $k_{0} ⩾ 0$ such that the solutions $u_{k}$ of the truncated problems (2.1) satisfy $\begin{array}{l} (6.11) & u_{k} \equiv u on [0, 1] \times [0, t_{1}] for all k ⩾ k_{0} . \end{array}$ Moreover, for any integer $j ⩾ a_{1}$ , there exists $k_{j} ⩾ max (k_{0}, j)$ such that $U_{j}$ is a steady state of problem (2.1) with $k = k_{j}$ . Since $u_{k_{j}}$ and $U_{j}$ are smooth, with $u_{k_{j}} = U_{j} = 0$ at $x = 0$ and $u_{k_{j}} = 0 < U_{j}$ at $x = 1$ , we can apply the zero number principle to infer that $\begin{array}{l} z (u_{k_{j}} (\cdot, t) - U_{j}) ⩽ z (u_{k_{j}} (\cdot, t_{1}) - U_{j}) . \end{array}$ This combined with (6.10), (6.11) yields $\begin{array}{l} z (u_{k_{j}} (\cdot, t) - U_{j}) ⩽ z (u (\cdot, t_{1}) - U_{j}) ⩽ N (t_{0}) . \end{array}$ Since $u_{k_{j}} (\cdot, t) - U_{j} \to u (\cdot, t) - U^{}$ in $C_{loc} (0, 1)$ as $j \to \infty$ , property (6.1) follows from Lemma 6.4. The assertion is proved. □

In view of the proof of assertions (ii)(iii), we prepare the following lemma which gives a more general property of the zero number on intervals $[0, b]$ and will be useful also in the proof of Theorem 1.3. In what follows, for any $b \in (0, 1]$ , we denote $\begin{array}{l} N_{b} (t) : = z_{| [0, b]} (u (\cdot, t) - U^{}), t > 0 . \end{array}$ Lemma 6.6.
Let $ϕ \in X_{1}$ , $b \in (0, 1]$ and $τ_{2} > τ_{1} > 0$ be such that $\begin{array}{l} (6.12) & u - U^{} \neq 0 on {b} \times [τ_{1}, τ_{2}] . \end{array}$ Then the function* $N_{b} (t)$ is nonincreasing and right continuous on $[τ_{1}, τ_{2}]$ .
Proof.
It suffices to consider the case $b \in (0, 1)$ . Indeed if $b = 1$ then, by continuity, for $η \in (0, 1)$ small, assumption (6.12) remains true for $b = 1 - η$ and we have $N_{1} (t) = z_{| [0, 1 - η]} (u (\cdot, t) - U^{})$ for all $t \in [τ_{1}, τ_{2}]$ .

Step 1. Upper semicontinuity on the right.*

We claim that for each $t \in [τ_{1}, τ_{2})$ , $\begin{array}{l} (6.13) & \underset{s \to t^{+}}{lim sup} N_{b} (s) ⩽ N_{b} (t) . \end{array}$ Set $w = u - U^{}$ . Since $0 ⩽ N (t) < \infty$ by Proposition 6.1(i), there exist $σ \in {- 1, 1}$ and $a \in (0, 1)$ such that $\begin{array}{l} σ w (\cdot, t) ⩾ 0 on [0, a] and σ w (a, t) > 0 . \end{array}$ By continuity, there exists $ε \in (0, τ_{2} - t)$ such that $σ w (a, s) > 0$ for all $s \in [t, t + ε]$ . Applying Proposition 2.1 if $σ = 1$ and Proposition 2.2 if $σ = - 1$ , we then have $\begin{array}{l} (6.14) & σ w (\cdot, t) ⩾ 0 on [0, a] \times [t, t + ε] and σ w (a, t) > 0 on [t, t + ε] . \end{array}$ We may suppose $b > a$ , since otherwise $N_{b} = 0$ on $[t, t + ε]$ and we are done. By (6.14) we have $\begin{array}{l} (6.15) & N_{b} (s) = z_{| [a, b]} (u (\cdot, s) - U^{}), t ⩽ s ⩽ t + ε . \end{array}$ Since $u, U^{}$ are smooth in $[a, b] \times [t, t + ε]$ and $u \neq U^{}$ on ${a, b} \times [t, t + ε]$ due to (6.12), (6.14), we may apply the zero number principle on $[a, b]$ . In view of (6.15) this yields $N_{b} (s) ⩽ N_{b} (t)$ for all $s \in [t, t + ε]$ , hence (6.13).

Step 2. Monotonicity in $[τ_{0}, τ_{1}]$ . Let $τ_{1} ⩽ t_{1} < t_{2} ⩽ τ_{2}$ and suppose for contradiction that $\begin{array}{l} (6.16) & N_{b} (t_{2}) ⩾ N_{b} (t_{1}) + 1 . \end{array}$ Set $t_{3} : = inf {t \in (t_{1}, t_{2}]; N_{b} (t) ⩾ N_{b} (t_{1}) + 1} \in [t_{1}, t_{2}]$ . We claim that $\begin{array}{l} (6.17) & N_{b} (t_{3}) ⩽ N_{b} (t_{1}) . \end{array}$ Indeed this is obvious if $t_{3} = t_{1}$ whereas, if $t_{3} > t_{1}$ , then $N_{b} (t) ⩽ N_{b} (t_{1})$ on $[t_{1}, t_{3})$ , so that Lemma 6.4 implies (6.17). By (6.16), (6.17) we thus have $t_{3} < t_{2}$ , hence there exists a sequence $t_{j} \to t_{3}^{+}$ such that $N_{b} (t_{j}) ⩾ N_{b} (t_{1}) + 1$ . But this contradicts (6.13). Consequently, $N_{b}$ is nonincreasing on $[τ_{0}, τ_{1}]$ . □
Proof of Proposition 6.1(ii)(iii).
The first part of assertion (ii) is a direct consequence of Lemma 6.6. The second part follows from the fact that, by the strong maximum principle, $\begin{array}{l} (6.18) & ‖ ϕ ‖_{\infty} ⩽ c_{p} ⟹ {‖ u (t) ‖}_{\infty} < c_{p} for all t > 0 . \end{array}$

To prove assertion (iii), taking $t \in T$ , we have $u \neq c_{p}$ on ${0} \times [t - ε, t + ε]$ for $ε > 0$ small by continuity. It thus follows from assertion (ii) that N is nonincreasing and right continuous on $[t - ε, t + ε]$ . Therefore $N (t^{\pm})$ exist and (6.2) is true. □
Proof of Proposition 6.2.
Let $t_{0} \in (0, T_{0})$ , with $T_{0} : = min (T (ϕ_{1}), T (ϕ_{2}))$ , or $t_{0} = 0$ if $N (ϕ_{1} - ϕ_{2}) < \infty$ . Let $u_{k}, v_{k}$ be the global classical solutions of the truncated problems (2.1) with $Ω = (0, 1)$ , $ϕ = ϕ_{1}$ and $Ω = (0, b)$ , $ϕ = ϕ_{2}$ , respectively. Since ${sup}_{s \in [0, t_{0}]} (‖ u_{x} (s) ‖_{\infty} + ‖ v_{x} (s) ‖_{\infty}) < \infty$ , there exists $k_{0} ⩾ 0$ such that $\begin{array}{l} (6.19) & u_{k} \equiv u on [0, 1] \times [0, t_{0}] and v_{k} \equiv v on [0, b] \times [0, t_{0}], for all k ⩾ k_{0} . \end{array}$ The solutions $u_{k}, v_{k}$ are smooth, with $u_{k} = v_{k} = 0$ at $x = 0$ , $u_{k} = v_{k} = 0$ at $x = 1$ if $b = 1$ and $u_{k} > 0 = v_{k}$ at $x = b$ if $b \in (0, 1)$ . Consequently, we can apply the zero number principle to infer that $\begin{array}{l} z_{| [0, b]} ((u_{k} - v_{k}) (\cdot, t)) ⩽ z_{| [0, b]} ((u_{k} - v_{k}) (\cdot, t_{0})) = N (t_{0}) < \infty, t > t_{0} \end{array}$ (where the equality is due to (6.19)). Since $(u_{k} - v_{k}) (\cdot, t) \to (u - v) (\cdot, t)$ in $C_{loc} (0, 1)$ as $k \to \infty$ , we deduce from Lemma 6.4 and assumption (6.4) that $N (t)$ is finite (i.e., $N (t) \in N$ ) for all $t ⩾ t_{0}$ .

We next claim that for each $t ⩾ t_{0}$ , $\begin{array}{l} (6.20) & \underset{s \to t^{+}}{lim sup} N (s) ⩽ N (t) . \end{array}$ Set $w = u - v$ . Since $N (t)$ is finite, by the argument in Step 1 of the proof of Lemma 6.6, it follows that there exist $a_{0} \in (0, 1)$ , $ε_{0} > 0$ and $σ_{0} \in {- 1, 1}$ such that $\begin{array}{l} (6.21) & σ_{0} w (\cdot, t) ⩾ 0 on [0, a_{0}] \times [t, t + ε_{0}] and σ_{0} w (a_{0}, t) > 0 on [t, t + ε_{0}] . \end{array}$ If $b = 1$ , then we similarly find $a_{1} \in (0, 1)$ , $ε_{1} > 0$ and $σ_{1} \in {- 1, 1}$ such that $\begin{array}{l} (6.22) & σ_{1} w (\cdot, t) ⩾ 0 on [a_{1}, 1] \times [t, t + ε_{1}] and σ_{1} w (a_{1}, t) > 0 on [t, t + ε_{1}] . \end{array}$ If $b \in (0, 1)$ then, in view of assumption (6.4) and since $u ⩾ e^{t Δ} ϕ > 0$ in $(0, 1) \times (0, \infty)$ , there exist $a_{1} \in (0, b)$ , $ε_{1} > 0$ such that $\begin{array}{l} (6.23) & w (\cdot, t) > 0 on [a_{1}, b] \times [t, t + ε_{1}] . \end{array}$ Letting $ε = min (ε_{0}, ε_{1})$ we may suppose $a_{1} > a_{0}$ , since otherwise $N = 0$ on $[t, t + ε]$ and we are done. By (6.21)–(6.23) we have $\begin{array}{l} (6.24) & N (s) = z_{| [a_{0}, a_{1}]} (w (\cdot, s)), t ⩽ s ⩽ t + ε . \end{array}$ Since $u, v$ are smooth on $[a_{0}, a_{1}]$ and $u \neq v$ on ${a_{0}, a_{1}} \times [t, t + ε]$ due to (6.21)–(6.23), we may apply the zero number principle on $[a_{0}, a_{1}]$ . In view of (6.24) this yields $N (s) ⩽ N (t)$ for all $s \in [t, t + ε]$ , hence (6.20).

The monotonicity of $N$ then follows exactly as in Step 2 of the proof of Lemma 6.6. □
Proof of Proposition 6.3.
Fix $ϕ_{0} \in X_{1}$ , with $\begin{array}{l} (6.25) & ϕ_{0} symmetric and nondecreasing on [0, \frac{1}{2}], \end{array}$ such that the corresponding solution v of (1.1) satisfies $v (0, t_{1}) = v (1, t_{1}) > 0$ for some $t_{1} > 0$ . Let $ϕ = λ ϕ_{0}$ with $λ > 1$ . Setting $μ = λ^{- 1}$ and $z = μ u$ , we have $\begin{array}{l} z_{t} - z_{x x} - | z_{x} |^{p} = μ (u_{t} - u_{x x} - μ^{p - 1} | u_{x} |^{p}) = μ (1 - μ^{p - 1}) | u_{x} |^{p} ⩾ 0 in (0, 1) \times (0, \infty) . \end{array}$ It thus follows from Proposition 2.2 that $z ⩾ v$ , hence $u ⩾ λ v$ in $[0, 1] \times [0, \infty)$ .

Now taking $λ > max {1, {(v (1, t_{1}))}^{- 1} c_{p}}$ , we get $u (1, t_{1}) > c_{p}$ . Also, assumption (6.25) guarantees that $u (\cdot, t)$ is symmetric and nondecreasing on $[0, \frac{1}{2}]$ for each $t > 0$ (arguing on the truncated problems (2.1) and passing to the limit). Therefore, $u (\cdot, t_{1}) > c_{p} ⩾ U^{*}$ in $[0, 1]$ , hence $N (t_{1}) = 0$ . But we know from [35] that u eventually decays to 0 in $C^{1}$ norm, hence $N (t) = 0$ for $t ≫ 1$ . This implies that, at some $t_{2} > t_{1}$ , $0 < u (0, t_{2}) = u (1, t_{2}) < c_{p}$ , hence $N (t_{2}) ⩾ 1$ . □

7. Zero number properties II: Analysis of the transition set $T$

Define the sets: $\begin{array}{l} Σ_{+} = {t > 0; \exists a \in (0, 1), u (\cdot, t) ⩾ U^{*} on (0, a) and u (a, t) > U^{*} (a)}, \\ Σ_{-} = {t > 0; \exists a \in (0, 1), u (\cdot, t) ⩽ U^{*} on (0, a) and u (a, t) < U^{*} (a)} . \end{array}$ We have $Σ_{+} \cap Σ_{-} = \emptyset$ and, as a consequence of Proposition 6.1(i), $\begin{array}{l} (7.1) & (0, \infty) = Σ_{+} \cup Σ_{-} . \end{array}$ Moreover, we have $\begin{array}{l} (7.2) & Σ_{+} \subset S \end{array}$ (this is obvious if $u (0, t) = 0$ and follows from Lemma 5.1(ii) if $u (0, t) > 0$ ).

Next, for all $t_{0} \in (0, T (ϕ))$ , we have $\begin{array}{l} (7.3) & z (u_{t} (\cdot, t)) ⩽ z (u_{t} (\cdot, t_{0})) < \infty, t > t_{0} . \end{array}$ Indeed, since u is smooth on $[0, 1] \times (0, T (ϕ))$ , we have $z (u_{t} (\cdot, t_{0})) < \infty$ by standard properties of the zero number applied to the equation for $u_{t}$ , and (7.3) then follows from [34, Proposition 6.2].

The goal of this section is to prove the following two propositions. The first one guarantees immediate loss of boundary conditions (resp., regularization) at any time at which the solution dominates (resp., is dominated by) the singular steady state near the boundary. Moreover, the recovery of boundary conditions or loss of regularity cannot occur as long as the intersection number remains constant.

Proposition 7.1.
Let $ϕ \in X_{1}$ , $b \in (0, 1]$ and $t_{2} > t_{1} > 0$ be such that $\begin{array}{l} (7.4) & N_{b} (t) = z_{| [0, b]} (u (\cdot, t) - U^{}) is constant on [t_{1}, t_{2}] . \end{array}$

(i) If* $t_{1} \in Σ_{+}$ , then $u > 0$ on ${0} \times (t_{1}, t_{2}]$ .

(ii) If $t_{1} \in Σ_{-}$ , then $u = 0$ on ${0} \times (t_{1}, t_{2}]$ and u is a classical solution on $[0, \frac{1}{2}] \times (t_{1}, t_{2}]$ .

Our second proposition shows that the intersection number has to drop at any transition time $t \in T$ , provided no intersection occurs at $x = 1$ . We actually need a slightly more general version on intervals $[0, b] \subset [0, 1]$ .
Proposition 7.2.
Let $ϕ \in X_{1}$ , $b \in (0, 1]$ and assume that $t \in T$ satisfies $u (b, t) \neq U^{} (b)$ . Then* $N_{b} (t^{-})$ is well defined and we have $\begin{array}{l} (7.5) & N_{b} (t^{-}) ⩾ N_{b} (t) + 1 . \end{array}$ Moreover, if $‖ ϕ ‖_{\infty} ⩽ c_{p}$ or if ϕ is symmetric, then ( 7.5 ) with $b = 1$ is true for any $t \in T$ .

In view of the proof of Propositions 7.1 and 7.2 we need the following two lemmas.
Lemma 7.3.
Let $ϕ \in X_{1}$ , $b \in (0, 1]$ and $t_{2} > t_{1} > 0$ satisfy ( 7.4 ). Then there exists $a > 0$ such that $\begin{array}{l} u > U^{} in (0, a] \times (t_{1}, t_{2}] \end{array}$ or $\begin{array}{l} u < U^{} in (0, a] \times (t_{1}, t_{2}] . \end{array}$
Proof.
Set $w = u - U^{}$ . We first note that, for any $t_{1} ⩽ s < t ⩽ t_{2}$ , $d \in (0, 1)$ and $σ \in {- 1, 1}$ , $\begin{array}{l} (7.6) & \begin{aligned} σ w ⩽ 0 in [0, d] \times [s, t] \\ σ w < 0 on {d} \times (s, t] \end{aligned}} ⟹ σ w < 0 in (0, d] \times (s, t], \end{array}$ as a consequence of the strong maximum principle (applied on each interval $[η, d]$ with $η \in (0, d)$ , where u and $U^{}$ are smooth).

Next, by (7.1), there exist $c \in (0, 1)$ and $\bar{σ} \in {- 1, 1}$ such that $\bar{σ} w (\cdot, t_{1}) ⩽ 0$ in $[0, c]$ and $\bar{σ} w (c, t_{1}) < 0$ . By continuity, there exists $t_{3} \in (t_{1}, t_{2})$ such that $\bar{σ} w (c, t) < 0$ for all $t \in [t_{1}, t_{3}]$ . Applying Proposition 2.1 if $\bar{σ} = - 1$ and Proposition 2.2 if $\bar{σ} = 1$ , we deduce that $\bar{σ} w ⩽ 0$ in $[0, c] \times [t_{1}, t_{3}]$ , hence $\bar{σ} w < 0$ in $(0, c] \times (t_{1}, t_{3}]$ , by (7.6).

Now, the set $\begin{array}{l} J : = {t \in (t_{1}, t_{2}]; \bar{σ} w < 0 in (0, d] \times (t_{1}, t] for some d > 0} \end{array}$ is a nontempty interval and we put $τ = sup J \in (t_{1}, t_{2}]$ . Since $N_{b} (τ) = m : = N_{b} (t_{1})$ , there exist $0 < x_{1} < \dots < x_{m + 1} < b$ and $\hat{σ} \in {- 1, 1}$ such that $\begin{array}{l} {(- 1)}^{i} \hat{σ} w (x_{i}, τ) > 0, i = 1, \dots, m + 1 . \end{array}$ By continuity, there exists $ε \in (0, τ - t_{1})$ such that $\begin{array}{l} (7.7) & {(- 1)}^{i} \hat{σ} w (x_{i}, t) > 0, i = 1, \dots, m + 1, for all t \in [τ - ε, \bar{τ}], \end{array}$ where $\bar{τ} = min (t_{2}, τ + ε)$ . It follows from (7.7) and $N_{b} (t) = m$ that $\begin{array}{l} \hat{σ} w ⩽ 0 in [0, x_{1}] \times [τ - ε, \bar{τ}] and \hat{σ} w < 0 on {x_{1}} \times [τ - ε, \bar{τ}], \end{array}$ hence $\begin{array}{l} (7.8) & \hat{σ} w < 0 in (0, x_{1}] \times (τ - ε, \bar{τ}] \end{array}$ by (7.6). Now, since $τ - ε \in J$ , there exists $a \in (0, x_{1})$ such that $\bar{σ} w < 0$ in $(0, a] \times (t_{1}, τ - ε]$ . By (7.8) and continuity, we deduce that $\hat{σ} = \bar{σ}$ , hence $\begin{array}{l} (7.9) & \bar{σ} w < 0 in (0, a] \times (t_{1}, \bar{τ}] . \end{array}$ Therefore, $\bar{τ} \in J$ hence $\bar{τ} ⩽ τ$ , i.e. $\bar{τ} = t_{2}$ , so that (7.9) proves the lemma. □
Lemma 7.4.
Let $ϕ \in X_{1}$ , $b \in (0, 1]$ and let $t_{2} > t_{1} > 0$ satisfy ( 7.4 ).

(i) If $u > 0$ on ${0} \times [t_{1}, t_{2})$ , then $u (0, t_{2}) > 0$ .

(ii) If $u (0, t_{1}) = 0$ and $u_{x} (0, t_{1}) < \infty$ , then $u = 0$ on ${0} \times (t_{1}, t_{2}]$ and u is a classical solution on $[0, \frac{1}{2}] \times (t_{1}, t_{2}]$ .
Proof.
(i) By Lemma 7.3 there exists $a > 0$ such that $\begin{array}{l} (7.10) & u > U^{} in (0, a] \times (t_{1}, t_{2}] . \end{array}$ For $λ \in (0, 1)$ , let $u_{λ}$ be the global viscosity solution of problem (1.1) with initial data $λ ϕ$ . As a consequence of Proposition 2.2, we have $\begin{array}{l} (7.11) & {‖ u (\cdot, t) - u_{λ} (\cdot, t) ‖}_{\infty} ⩽ {‖ u (\cdot, 0) - u_{λ} (\cdot, 0) ‖}_{\infty} = (1 - λ) ‖ ϕ ‖_{\infty}, t > 0 \end{array}$ (this follows by taking $w = u + ‖ u (\cdot, 0) - u_{λ} (\cdot, 0) ‖_{\infty}$ as comparison function, and then exchanging the roles of u and $u_{λ}$ ). Since $z : = λ u$ satisfies $\begin{array}{l} z_{t} - z_{x x} - | z_{x} |^{p} = λ (u_{t} - u_{x x} - λ^{p - 1} | u_{x} |^{p}) = λ (1 - λ^{p - 1}) | u_{x} |^{p} ⩾ 0 in (0, 1) \times (0, \infty) \end{array}$ it follows from Proposition 2.2 that $\begin{array}{l} (7.12) & u_{λ} ⩽ λ u in (0, 1) \times (0, \infty) . \end{array}$

Fix $t_{3} \in (t_{1}, t_{2})$ and define the compact set $Γ = ([0, a] \times {t_{3}}) \cup ({a} \times [t_{3}, t_{2}])$ . Owing to (7.10) and $u (0, t_{3}) > 0$ , we have $η = {min}_{Γ} (u - U^{}) > 0$ . Taking λ close enough to 1, so that $(1 - λ) ‖ ϕ ‖_{\infty} < η$ and using (7.11), we get $\begin{array}{l} u_{λ} - U^{} ⩾ 0 on Γ . \end{array}$ Since $u_{λ} ⩾ 0$ on ${0} \times [t_{3}, t_{2}]$ it follows from Proposition 2.1 that $\begin{array}{l} u_{λ} ⩾ U^{} in (0, a) \times (t_{3}, t_{2}) . \end{array}$ Combining this with (7.12), we obtain $\begin{array}{l} u (\cdot, t_{2}) ⩾ λ^{- 1} u_{λ} (\cdot, t_{2}) ⩾ λ^{- 1} U^{} in (0, a) . \end{array}$ By Lemma 5.1(ii), we conclude that $u (0, t_{2}) > 0$ .

(ii) By our assumptions and (1.5), there exist $c \in (0, 1)$ and $μ_{0} > 0$ such that, for all $μ ⩾ μ_{0}$ , $\begin{array}{l} (7.13) & u (\cdot, t_{1}) < U_{μ} < U^{} in (0, c), \end{array}$ where the regular steady state $U_{μ}$ is defined in (6.3). Consequently, by Lemma 7.3, there exists $a \in (0, c)$ such that $\begin{array}{l} u < U^{} in (0, a] \times [t_{1}, t_{2}], \end{array}$ hence in particular $u = 0$ on ${0} \times (t_{1}, t_{2})$ . We may thus choose $λ > μ_{0}$ sufficiently large so that $\begin{array}{l} (7.14) & u ⩽ U_{λ} on {0, a} \times [t_{1}, t_{2}] . \end{array}$ It then follows from (7.13), (7.14) and Proposition 2.1 that $\begin{array}{l} u (x, t) < U_{λ} (x) ⩽ λ x for all (x, t) \in (0, a] \times [t_{1}, t_{2}] . \end{array}$ Let $(x, t) \in (0, a] \times [t_{1}, t_{2}]$ . By the mean-value theorem there exists $y \in (0, x)$ such that $u_{x} (y, t) = x^{- 1} u (x, t) ⩽ λ$ . Since $s \mapsto u_{x} (s, t) - K s$ is nonincreasing by (5.1) in Lemma 5.1(i) (applied with $t_{0} = t_{1}$ ), it follows that $u_{x} (x, t) - K x ⩽ u_{x} (y, t) - K y ⩽ λ$ , hence $u_{x} (x, t) ⩽ λ + K$ . This combined with (5.2)–(5.3) guarantees that $\begin{array}{l} sup_{(0, \frac{3}{4}) \times (t_{1}, t_{2})} | u_{x} | < \infty . \end{array}$ By parabolic estimates we conclude that u is a classical solution on $[0, \frac{1}{2}] \times (t_{1}, t_{2}]$ . □

With Lemmas 7.3 and 7.4 at hand we can now turn to the proof of Propositions 7.1 and 7.2.
Proof of Proposition 7.1.
Set $v = u - U^{}$ .

(i) Assume for contradiction that $u (0, t_{3}) = 0$ for some $t_{3} \in (t_{1}, t_{2}]$ . We consider the following two cases.

Case 1: there exists $t_{4} \in (t_{1}, t_{3})$ such that $u (0, t_{4}) > 0$ . Letting $\begin{array}{l} t_{5} = inf {s > t_{4}; u (0, s) = 0} \in (t_{4}, t_{3}], \end{array}$ we then have $u (0, t_{5}) = 0$ by continuity, whereas $u > 0$ on ${0} \times (t_{4}, t_{5})$ . But this contradicts Lemma 7.4(i).

Case 2: $\begin{array}{l} (7.15) & u (0, s) = 0 for all s \in (t_{1}, t_{3}] . \end{array}$ By our assumption, there exists $a \in (0, 1)$ such that $\begin{array}{l} (7.16) & v (\cdot, t_{1}) ⩾ 0 in [0, a] and v (a, t_{1}) > 0 . \end{array}$ By continuity there exists $t_{4} \in (t_{1}, t_{2})$ such that $v (a, s) > 0$ for all $s \in (t_{1}, t_{4}]$ . Applying the comparison principle (Proposition 2.1) and then the strong maximum principle, it follows that $\begin{array}{l} (7.17) & v > 0 in (0, a] \times (t_{1}, t_{4}] . \end{array}$ In particular (7.15) and (7.17) imply that $\begin{array}{l} (7.18) & u_{x} (0, s) = \infty for all s \in (t_{1}, t_{4}] . \end{array}$ Pick $t_{5} \in (t_{1}, t_{4})$ . By (7.3) we have $z (u_{t} (\cdot, t_{5})) < \infty$ . Consequently, we have either $u_{t} (\cdot, t_{5}) ⩾ 0$ or $u_{t} (\cdot, t_{5}) ⩽ 0$ for $x > 0$ small. We deduce from Lemma 5.2 and (7.18) that either $\begin{array}{l} (7.19) & u_{x} (\cdot, t_{5}) ⩾ U_{x}^{} or u_{x} (\cdot, t_{5}) ⩽ U_{x}^{} for x > 0 small. \end{array}$ The second alternative in (7.19) cannot hold in view of (7.15), since this would lead to $v (\cdot, t_{5}) ⩽ 0$ for $x > 0$ small, contradicting (7.17).

Therefore, there exists $c \in (0, a)$ such that $\begin{array}{l} u_{x} (\cdot, t_{5}) ⩾ U_{x}^{} in (0, c) \end{array}$ and $u_{x} (b, t_{5}) > U_{x}^{} (c)$ . By continuity, there exist $t_{6} \in (t_{5}, t_{4})$ such that $\begin{array}{l} u_{x} (c, s) > U_{x}^{} (c) for all s \in [t_{5}, t_{6}] . \end{array}$ Lemma 5.3(ii) and (7.18) then guarantee that $u_{x} ⩾ U_{x}^{}$ in $Q = : (0, c) \times (t_{5}, t_{6}]$ . Consequently, the function v satisfies $\begin{array}{l} v_{t} - v_{x x} = {(u_{x})}^{p} - {(U_{x}^{})}^{p} ⩾ 0 in Q, \end{array}$ with $v > 0$ in Q. By comparison with the solution of the heat equation, it follows that $v (\cdot, t_{6}) ⩾ ε x$ in $(0, c / 2)$ for some $ε > 0$ . But, recalling (7.15), this contradicts estimate (5.5) in Lemma 5.1. Assertion (i) is proved.

(ii) By our assumption, there exists $a \in (0, 1)$ such that $\begin{array}{l} (7.20) & v (\cdot, t_{1}) ⩽ 0 in [0, a] and v (a, t_{1}) < 0 . \end{array}$ By continuity there exists $t_{3} \in (t_{1}, t_{2})$ such that $v (a, s) < 0$ for all $s \in [t_{1}, t_{3}]$ . It follows from Proposition 2.2 that $u ⩽ U^{}$ in $[0, a] \times [t_{1}, t_{3}]$ , hence in particular $\begin{array}{l} (7.21) & u (0, s) = 0 for all s \in [t_{1}, t_{3}] . \end{array}$ Moreover, by the strong maximum principle, we get $\begin{array}{l} (7.22) & v < 0 in (0, a] \times (t_{1}, t_{3}] . \end{array}$ We consider the following two cases.

Case 1: there exists a sequence $s_{i} \to t_{1}$ with $s_{i} > t_{1}$ , such that $u_{x} (0, s_{i}) < \infty$ . By Lemma 7.4(ii), it follows that u is a classical solution on $[0, \frac{1}{2}] \times (s_{i}, t_{2}]$ for each i, hence on $[0, \frac{1}{2}] \times (t_{1}, t_{2}]$ , which is the desired result.

Case 2: there exists $t_{4} \in (t_{1}, t_{3})$ such that $\begin{array}{l} (7.23) & u_{x} (0, s) = \infty for all s \in (t_{1}, t_{4}] . \end{array}$ Pick $t_{5} \in (t_{1}, t_{4})$ . By (7.3) we have $z (u_{t} (\cdot, t_{5})) < \infty$ . Consequently, we have either $u_{t} (\cdot, t_{5}) ⩾ 0$ or $u_{t} (\cdot, t_{5}) ⩽ 0$ for $x > 0$ small. We deduce from Lemma 5.2 and (7.23) that either $\begin{array}{l} (7.24) & u_{x} (\cdot, t_{5}) ⩾ U_{x}^{} or u_{x} (\cdot, t_{5}) ⩽ U_{x}^{} for x > 0 small. \end{array}$ The first alternative in (7.24) cannot hold, since this would lead to $v (\cdot, t_{5}) ⩾ 0$ for $x > 0$ small, contradicting (7.22).

Let $\bar{x}$ be given by Lemma 5.1(i) with $t_{0} = t_{1}$ . Then there exists $c \in (0, min (\bar{x}, a))$ such that $\begin{array}{l} u_{x} (\cdot, t_{5}) ⩽ U_{x}^{} in (0, c) and u_{x} (c, t_{5}) < U_{x}^{} (c) . \end{array}$ By continuity, there exists $t_{6} \in (t_{5}, t_{4})$ such that $\begin{array}{l} u_{x} (c, s) < U_{x}^{} (c) for all s \in [t_{5}, t_{6}] . \end{array}$ Lemma 5.3(i) then guarantees that $u_{x} ⩽ U_{x}^{}$ in $Q : = (0, c) \times (t_{5}, t_{6}]$ . Since also $u_{x} ⩾ - U_{x}^{}$ in Q by (5.4), the function v thus satisfies $\begin{array}{l} v_{t} - v_{x x} = | u_{x} |^{p} - {(U_{x}^{})}^{p} ⩽ 0 in Q, \end{array}$ with $v < 0$ in Q. By comparison with the solution of the heat equation, it follows that $v (\cdot, t_{6} ⩽ - ε x$ in $(0, c / 2)$ for some $ε > 0$ . But recalling (7.23), this contradicts estimate (5.5) in Lemma 5.1. The lemma is proved. □
Proof of Proposition 7.2.
By continuity, there exists $ε > 0$ such that $u (s, b) \neq U^{} (b)$ for all $s \in [t - ε, t + ε]$ and Lemma 6.6 thus guarantees that the function $N_{b} (t)$ is nonincreasing and right continuous on $[t - ε, t + ε]$ . Assume for contradiction that $N_{b} (t^{-}) = N_{b} (t)$ . Since $N_{b} (t)$ is integer valued, there exists $\hat{ε} \in (0, ε)$ such that $\begin{array}{l} (7.25) & N_{b} (s) = N_{b} (t) for all s \in [t - \hat{ε}, t + \hat{ε}] . \end{array}$ If $t - \hat{ε} \in Σ_{+}$ then, by Proposition 7.1(i), $u (0, s) > 0$ for all $s \in (t - ε, t + ε)$ , contradicting $u (0, t) = 0$ . If $t - \hat{ε} \in Σ_{-}$ then, by Proposition 7.1(ii), u is a classical solution on $[0, \frac{1}{2}] \times (t - \hat{ε}, t + \hat{ε})$ , contradicting $u_{x} (0, t) = \infty$ .

The last statement of the proposition follows from the fact that, when $‖ ϕ ‖_{\infty} ⩽ c_{p}$ or ϕ is symmetric, the condition $u (1, t) \neq U^{} (1)$ is satisfied for all $t \in T$ , owing respectively to (6.18) or to $u (1, t) = u (0, t) = 0$ . □

We end this section with the following variant of Lemma 7.3, concerning the number of intersections of two solutions, which will be useful in the proof of Theorem 1.4. Lemma 7.5.
Let $0 < b ⩽ 1$ , $ϕ_{1} \in X_{1}$ , $ϕ_{2} \in X_{b}$ , and denote by $u, v$ the corresponding viscosity solutions of ( 1.1 ), respectively with $Ω = (0, 1)$ and $Ω = (0, b)$ . Let $t_{2} > t_{1} > 0$ and assume that $\begin{array}{l} z_{| [0, b]} (u (\cdot, t) - v (\cdot, t)) is finite and constant on [t_{1}, t_{2}] . \end{array}$ Then there exists $a \in (0, b)$ such that $\begin{array}{l} u > v in (0, a] \times (t_{1}, t_{2}] \end{array}$ or $\begin{array}{l} u < v in (0, a] \times (t_{1}, t_{2}] . \end{array}$
Proof of Lemma 7.5.
It is completely similar to the proof of Lemma 7.3, replacing $U^{*}$ with v. □

8. Proof of Theorem 1.3

We first give a simpler proof of Theorem 1.3(i) in a special but already representative case.

Proof of Theorem 1.3(i) under the assumption $‖ ϕ ‖_{\infty} ⩽ c_{p}$ ..
Set $T = T (ϕ)$ and fix $t_{0} = 0$ if $N (0) < \infty$ , or any $t_{0} \in (0, T)$ otherwise. By Propositions 6.1(ii) and 7.2, $N (t)$ is nonincreasing on $[t_{0}, \infty)$ and we have $N (t^{-}) ⩾ N (t) + 1$ for all $t \in T$ . Since $T \cap [0, T) = \emptyset$ , it follows that $# T ⩽ N (t_{0})$ . This proves the assertion. □

We now turn to the proof of Theorem 1.3(i) in the general case. Since N need not be monotone in general (cf. Proposition 6.3), the proof is more delicate. We shall show that, around any transition time $t \in T$ , the number of intersections of the approximate solutions $u_{k}$ with regular steady states $U_{j}$ (cf. (6.3)) has to drop for suitably large $j, k$ . This is the purpose of the following proposition. In what follows, for any positive integers $j, k$ , we set $\begin{array}{l} Z_{k, j} (t) = z_{[0, 1]} (u_{k} (\cdot, t) - U_{j}), t ⩾ 0 . \end{array}$ Proposition 8.1.
Let $t \in T$ . For any $τ \in (0, t)$ , there exist $j_{0} ⩾ 1$ and a sequence of integers ${(κ_{j})}_{j ⩾ j_{0}}$ (depending on $t, τ$ ) such that, for all $j ⩾ j_{0}$ and $k ⩾ κ_{j}$ , $\begin{array}{l} (8.1) & Z_{k, j} is finite and nonincreasing on (0, \infty) \end{array}$ and $\begin{array}{l} (8.2) & Z_{k, j} (t - τ) ⩾ Z_{k, j} (t + τ) + 1 . \end{array}$
Proof.
We need to carefully control the zeros of $u_{k} - U_{j}$ and of $u - U^{}$ near $x = 0$ , near $x = 1$ , and on the remaining interval. We thus proceed in three steps.

Step 1. Behavior near* $x = 1$ . We consider two cases.

Case A. $u (1, t) < c_{p}$ . Then there exists $ε \in (0, τ)$ such that $u (1, s) < c_{p}$ for all $s \in [t - ε, t + ε]$ . We may thus choose $b \in (0, 1)$ and $j_{1} ⩾ 1$ such that $\begin{array}{l} (8.3) & u (\cdot, s) - U^{} < 0 in [b, 1] for all s \in [t - ε, t + ε] \end{array}$ and $u_{k} (\cdot, s) - U_{j} < 0$ in $[b, 1]$ for all $s \in [t, t + ε]$ , $j ⩾ j_{1}$ and $k ⩾ 1$ , hence $\begin{array}{l} (8.4) & Z_{k, j} (s) = z_{[0, b]} (u_{k} (\cdot, s) - U_{j}) for all s \in [t - ε, t + ε], j ⩾ j_{1} and k ⩾ 1 . \end{array}$

Case B.* $u (1, t) ⩾ c_{p}$ . As a consequence of Lemma 5.4 (applied to the reflected solution $\tilde{u} (x, t) = u (1 - x, t)$ ), there exist $b \in (0, 1)$ and $k_{1} ⩾ 1$ such that, for all $s > 0$ , $\begin{array}{l} u (1, s) > 0 & ⟹ \partial_{x} u_{k} (\cdot, s) ⩽ 0 on [b, 1) for all k ⩾ k_{1} \\ ⟹ u_{x} (\cdot, s) ⩽ 0 on [b, 1) . \end{array}$ We may thus choose $ε \in (0, τ)$ small enough so that $\begin{array}{l} (8.5) & u (b, s) ⩾ u (1, s) > U^{} (b) for all s \in [t - ε, t + ε] . \end{array}$ Since $u_{k} \to u$ locally uniformly on $(0, 1) \times [0, \infty)$ as $k \to \infty$ , there exists $k_{2} ⩾ k_{1}$ , such that for all $s \in [t - ε, t + ε]$ , $j ⩾ 1$ and $k ⩾ k_{2}$ , the function $u_{k} (\cdot, s) - U_{j}$ is decreasing on $[b, 1]$ , positive at $x = b$ and negative at $x = 1$ (recalling that $u_{k} (1, s) = 0$ ). Consequently $\begin{array}{l} (8.6) & Z_{k, j} (s) = z_{[0, b]} (u_{k} (\cdot, s) - U_{j}) + 1 for all s \in [t - ε, t + ε], j ⩾ 1 and k ⩾ k_{2} . \end{array}$

In each case, by Lemma 6.6 and Proposition 7.2 there exists $t_{1} \in (t, t + ε)$ such that $\begin{array}{l} (8.7) & N_{b} (σ) - 1 ⩾ N_{b} (t) = N_{b} (s) for t - ε ⩽ σ < t < s ⩽ t_{1} . \end{array}$

Step 2. Behavior near* $x = 0$ . We set $t_{2} = (t + t_{1}) / 2$ and consider the cases $t \in Σ_{-}$ and $t \in Σ_{+}$ separately.

Case 1. $t \in Σ_{-}$ . By Proposition 7.1(ii) and (8.7), u is a classical solution on $[0, \frac{1}{2}] \times (t, t_{1}]$ . Consequently, since ${sup}_{[0, \frac{1}{2}] \times [t_{2}, t_{1}]} | u_{x} | < \infty$ , there exist $a \in (0, b)$ and $j_{2} ⩾ j_{1}$ such that $\begin{array}{l} (8.8) & u < U_{j_{2}} < U^{} in (0, a] \times [t_{2}, t_{1}], \end{array}$ hence $\begin{array}{l} (8.9) & u_{k} (\cdot, s) - U_{j} < 0 in (0, a] for all s \in [t_{2}, t_{1}], j ⩾ j_{2} and k ⩾ 1 . \end{array}$

Case 2.* $t \in Σ_{+}$ . By Lemma 7.3 and (8.7) there exists $a \in (0, 1)$ such that $\begin{array}{l} (8.10) & u > U^{} in (0, a] \times (t, t_{1}] . \end{array}$ hence in particular $[t, t_{1}] \subset Σ_{+} \subset S$ by (7.2). We then deduce from Lemma 5.4 that for each $j ⩾ 1$ , there exist $η_{j} \in (0, a)$ and $κ_{1} (j) ⩾ k_{2}$ such that $\begin{array}{l} u_{k} > j x ⩾ U_{j} in (0, η_{j}] \times (t, t_{1}] for all k ⩾ κ_{1} (j) . \end{array}$ Since $u_{k} \to u$ locally uniformly on $(0, 1) \times [t, t_{1}]$ as $k \to \infty$ and $U > U_{j}$ on $(0, 1]$ , we infer from (8.10) the existence of $κ_{2} (j) ⩾ κ_{1} (j)$ such that $u_{k} > U_{j}$ in $[η_{j}, a] \times [t, t_{1}]$ for all $k ⩾ κ_{2} (j)$ . Therefore, we have $\begin{array}{l} (8.11) & u_{k} (\cdot, s) - U_{j} > 0 in (0, a] for all s \in [t, t_{1}], j ⩾ 1 and k ⩾ κ_{2} (j) . \end{array}$

Step 3. Conclusion.* For any integer $j ⩾ 1$ , there exists $κ_{3} (j) ⩾ κ_{2} (j)$ such that $U_{j}$ is a steady state of problem (2.1) for all $k ⩾ κ_{3} (j)$ . Since $u_{k}, U_{j}$ are smooth, $u_{k} = U_{j} = 0$ at $x = 0$ and $u_{k} = 0 \neq U_{j}$ at $x = 1$ , it follows from the zero number principle that $\begin{array}{l} (8.12) & Z_{k, j} is finite and nonincreasing in (0, \infty) for all j ⩾ 1 and k ⩾ κ_{3} (j) . \end{array}$ Also, since u and $U^{}$ are classical solutions of $u_{t} - u_{x x} = | u_{x} |^{p}$ on $[a, b]$ , in view of (8.3), (8.5), (8.8), (8.10), the function $z_{[a, b]} (u (\cdot, t) - U^{})$ is finite and nonincreasing on $(t_{2}, t_{1})$ and it drops at any time s such that $u (\cdot, s) - U^{}$ has a degenerate zero. Consequently, there exist $\hat{t} \in (t_{2}, t_{1})$ such that all zeros of $u (\cdot, \hat{t}) - U$ on $[a, b]$ are nondegenerate. Since $u_{k} (\cdot, \hat{t}) - U_{j}$ converges to $u (\cdot, \hat{t}) - U^{}$ in $C^{1} ([a, b])$ as $k, j \to \infty$ , there exists $k_{3} ⩾ max (j_{2}, k_{2})$ such that $\begin{array}{l} z_{[a, b]} (u_{k} (\cdot, \hat{t}) - U_{j}) = z_{[a, b]} (u (\cdot, \hat{t}) - U^{}) for all k, j ⩾ k_{3} . \end{array}$ For any $j ⩾ k_{3}$ and $k ⩾ max (k_{3}, κ_{3} (j))$ , using (8.8)–(8.11), we obtain $\begin{array}{l} (8.13) & z_{[0, b]} (u_{k} (\cdot, \hat{t}) - U_{j}) = z_{[0, b]} (u (\cdot, \hat{t}) - U^{}) = N_{b} (\hat{t}) . \end{array}$ On the other hand, by Lemma 6.4, there exists $j_{0} ⩾ k_{3}$ such that $z_{[0, b]} (u_{k} (\cdot, t - ε) - U_{j}) ⩾ N_{b} (t - ε)$ for all $k, j ⩾ j_{0}$ . For any $j ⩾ j_{0}$ and $k ⩾ κ_{j} : = max (j_{0}, κ_{3} (j))$ , this along with (8.7), (8.13) guarantees that $\begin{array}{l} (8.14) & z_{[0, b]} (u_{k} (\cdot, t - ε) - U_{j}) ⩾ N_{b} (t) + 1 ⩾ N_{b} (\hat{t}) + 1 = z_{[0, b]} (u_{k} (\cdot, \hat{t}) - U_{j}) + 1 . \end{array}$ In case A, by (8.4), (8.12), we get $\begin{array}{l} Z_{k, j} (t - ε) ⩾ Z_{k, j} (\hat{t}) + 1 ⩾ Z_{k, j} (t + ε) + 1 . \end{array}$ In case B, by (8.6) and (8.12), (8.14) yields $\begin{array}{l} Z_{k, j} (t - ε) - 1 & = z_{[0, b]} (u_{k} (\cdot, t - ε) - U_{j}) \\ ⩾ z_{[0, b]} (u_{k} (\cdot, \hat{t}) - U_{j}) + 1 = Z_{k, j} (\hat{t}) ⩾ Z_{k, j} (t + ε) . \end{array}$ In view of (8.12) and $ε \in (0, τ)$ , this proves the proposition. □
Proof of Theorem 1.3(i) in the general case.
Set $T = T (ϕ)$ and fix $τ_{0} = 0$ if $N (0) < \infty$ , or any $τ_{0} \in (0, T)$ otherwise. Pick $t_{0} \in (τ_{0}, T)$ . Then there exists $k_{0} ⩾ 1$ such that $u_{k} = u$ on $[0, 1] \times [0, t_{0}]$ for all $k ⩾ k_{0}$ . By Lemma 6.5, we deduce the existence of $k_{1} ⩾ k_{0}$ such that $\begin{array}{l} (8.15) & Z_{k, j} (t_{0}) = z (u (\cdot, t_{0}) - U_{j}) ⩽ N (τ_{0}) for all k, j ⩾ k_{1} . \end{array}$

Let $m ⩾ 1$ and assume that $t_{1}, \dots, t_{m} \in T \subset [T, \infty)$ , with $t_{1} < \dots < t_{m}$ if $m ⩾ 2$ . Set $τ = \frac{1}{2} {min}_{1 ⩽ i ⩽ m} (t_{i} - t_{i - 1})$ . For $1 ⩽ i ⩽ m$ , let $j_{0, i} ⩾ 1$ and ${(κ_{j}^{i})}_{j ⩾ j_{0, i}}$ be given by Proposition 8.1 applied with $t = t_{i}$ . Choose $\begin{array}{l} j = max (k_{1}, j_{0, 1}, \dots, j_{0, m}), k = max (k_{1}, κ_{j}^{1}, \dots, κ_{j}^{m}) \end{array}$ and set $s_{i} = (t_{i} + t_{i - 1}) / 2$ for all $i \in {1, \dots, m}$ and $s_{m + 1} = t_{m} + τ$ . Then it follows from (8.1)–(8.2) that $\begin{array}{l} Z_{k, j} (s_{i}) - Z_{k, j} (s_{i + 1}) ⩾ Z_{k, j} (t_{i} - τ) - Z_{k, j} (t_{i} + τ) ⩾ 1, i = 1, \dots, m, \end{array}$ hence $m ⩽ Z_{k, j} (s_{1}) - Z_{k, j} (s_{m + 1}) ⩽ Z_{k, j} (t_{0}) ⩽ N (τ_{0})$ by (8.15). This proves the assertion. □
Proof of Theorem 1.3(ii).
Let $t_{1}, t_{2} \in T$ be such that $t_{1} < t_{2}$ and $(t_{1}, t_{2}) \cap T = \emptyset$ . We consider two cases.

Case 1: there exists $t \in (t_{1}, t_{2})$ such that $u (0, t) > 0$ . Let $\begin{array}{l} τ : = min {s > t; u (0, s) = 0} \in (t, t_{2}] . \end{array}$ Then $u (0, s) > 0$ for all $s \in (t, τ)$ and, by Lemma 5.1(ii), we have $\begin{array}{l} | u (x, s) - u (0, s) - U^{} (x) | ⩽ K x^{2}, 0 < x ⩽ \frac{1}{2} . \end{array}$ Letting $s \to τ$ and using the continuity of u, we obtain $\begin{array}{l} | u (x, τ) - U^{} (x) | ⩽ K x^{2}, 0 < x ⩽ \frac{1}{2} . \end{array}$ Therefore $u_{x} (0, τ) = \infty$ , hence $τ \in T$ . Since $(t_{1}, t_{2}) \cap T = \emptyset$ , it follows that $τ = t_{2}$ . Similarly we obtain $max {s < t; u (0, s) = 0} = t_{1}$ , hence $u (0, \cdot) > 0$ on $(t_{1}, t_{2})$ .

Case 2: $u (0, t) = 0$ for all $t \in (t_{1}, t_{2})$ . We claim that $\begin{array}{l} (8.16) & sup_{(0, 3 / 4] \times [t_{1} + ε, t_{2} - ε]} | u_{x} | < \infty, for each ε > 0 . \end{array}$ If (8.16) fails, in view of (5.2), (5.3), there exist $ε > 0$ and sequences $t_{j} \in [t_{1} + ε, t_{2} - ε]$ and $x_{j} \to 0$ such that $u_{x} (x_{j}, t_{j}) \to \infty$ . Passing to a subsequence, we may assume that $t_{j} \to τ \in [t_{1} + ε, t_{2} - ε]$ . By Lemma 5.1(ii), it follows that $u_{x} (0, τ) = \infty$ , hence $τ \in T$ : a contradiction.

Finally, by (8.16) and parabolic estimates we conclude that u is a classical solution on $[0, \frac{1}{2}] \times (t_{1}, t_{2})$ . □

9. Proof of Theorem 1.4

It is convenient to first state and prove the following special case of Theorem 1.4, whose proof, based on Theorem 1.1 and on a simple application of zero number, is considerably easier but will serve as a starting point for the general case.

Theorem 9.1.

Let $p > 2$ , $n = 1$ , $Ω = (0, 1)$ and set $t_{0} = 0$ . For any integer $m ⩾ 1$ , there exist $ϕ \in X_{1}$ and times $0 < t_{1} < \dots < t_{2 m} \in T$ such that u is classical on $(t_{2 m}, \infty)$ and, for each $i \in {1, \dots, m}$ , $\begin{array}{l} u is classical on (t_{2 i - 2}, t_{2 i - 1}) and of LBC type on (t_{2 i - 1}, t_{2 i}) . \end{array}$

Proof of Theorem 9.1.

It follows from Theorem 1.1 that there exists $ϕ \in X$ such that u has at least m losses and recoveries of boundary conditions. Moreover, by inspection of the proof, one can see that ϕ can be taken in $X_{1}$ and chosen to have exactly $2 m$ intersections with $U^{*}$ . The conclusion then follows from Theorem 1.3. □

Solutions are described in Theorem 1.4 by arbitrary finite sequences ${(σ_{i})}_{1 ⩽ i ⩽ ℓ - 1}$ with values in ${C, L}$ . It is obvious that Theorem 1.4 can be equivalently reformulated as follows, and it will be more conveniently proved under this form.

Theorem 1.4’.

Let $n = 1$ , $Ω = (0, 1)$ and let d be a positive integer. For any finite sequence ${({\bar{σ}}_{i})}_{1 ⩽ i ⩽ d} \in N^{d}$ , there exist $ϕ \in X_{1}$ and times ${\bar{t}}_{d + 1} > \dots > {\bar{t}}_{1} > 0$ such that $\begin{array}{l} u (\cdot, t) is classical near x = 0 for t \in (0, {\bar{t}}_{1}) \cup ({\bar{t}}_{d + 1}, \infty) \\ and for t in the neighborhood of {\bar{t}}_{1}, \dots, {\bar{t}}_{d + 1} \end{array}$ and, for each $i \in {1, \dots, d}$ ,

∙ if ${\bar{σ}}_{i} = 0$ , then u is classical near $x = 0$ in $I_{i} : = ({\bar{t}}_{i}, {\bar{t}}_{i + 1})$ except for a single time;

∙ if ${\bar{σ}}_{i} ⩾ 1$ , then ${t \in I_{i}, u (0, t) > 0}$ is a nonempty open subinterval of $I_{i}$ minus ${\bar{σ}}_{i} - 1$ times.

Moreover, u is classical up to $x = 1$ for all times, i.e. $\begin{array}{l} (9.1) & u \in C^{2, 1} ([\frac{1}{2}, 1] \times (0, \infty)) and u = 0 on {1} \times (0, \infty) . \end{array}$

The integer ${\bar{σ}}_{i}$ represents the number of LBC bumps in the interval $I_{i}$ and the cases ${\bar{σ}}_{i} = 0, 1$ or $⩾ 2$ respectively correspond to the three possible behaviors within each interval $I_{i}$ , namely: GBU without LBC, single loss and recovery of boundary conditions, and bouncing (possibly multiple). The times ${\bar{t}}_{i}$ and indices ${\bar{σ}}_{i}$ , along with the transition times $t_{i}$ , are represented on Fig. 4 for a typical example.

Fig. 4.

The times ${\bar{t}}_{i}$ and indices ${\bar{σ}}_{i}$ (here $d = 3$ ).

Outline of proof of Theorem 1.4 / 1.4 ’. The proof is rather long and delicate. It is based on a modification of the proof of Theorems 1.1 and 9.1, along with deformation, zero number and recursion arguments.

In Theorem 9.1 (via Theorem 1.1), we have constructed multibump initial data such that the corresponding solutions satisfy $u (0, t) > 0$ on m open time intervals $J_{i}$ (LBC bumps), separated and surrounded by nontrivial closed intervals where $u (0, t) = 0$ . Moreover, our construction was made recursively, using a suitable rescaling for each bump, with the spatial bump closest to the boundary being responsible for the first LBC time interval $J_{1}$ of the solution, and so on.

An additional difficulty now is that, whereas multibump solutions with separated LBC time intervals should be rather stable, the phenomena of GBU without LBC or of bouncing are expected to be unstable. To produce an arbitrary solution as in the statements of Theorem 1.4/1.4’, the idea is to deform this multibump initial data by performing one of the following three operations on each space bump $B_{i}$ of the initial data, numbered from left to right by $i \in {1, \dots, m}$ :

Reduce the amplitude of $B_{i}$ by multiplying by a number in $(0, 1)$ , with aim of squeezing the corresponding LBC interval $J_{i}$ to a single time $t \in T$ . The resulting t produced in this way will be a GBU time without LBC, separating two intervals where u is classical;

(if $i ⩽ m - 1$ and operation (1) is not being made on $B_{i + 1}$ ) Link $B_{i}$ with its right neighbor $B_{i + 1}$ by a suitable deformation, with the aim of squeezing the separating interval between the LBC intervals $J_{i}$ , $J_{i + 1}$ to a single time $t \in T$ . The resulting t will be a bouncing time. Note that the operation (2) can be repeated with the next bump(s) on the right to create several consecutive rebounds;

Leave $B_{i}$ unchanged.

We will actually make a continuous deformation along operations (1) and (2) above, leading to a suitable q-parameter family of initial data $Φ_{μ_{1}, \dots, μ_{q}}$ with $(μ_{1}, \dots, μ_{q}) \in {[0, 1]}^{q}$ and the desired solution will then be obtained by iteratively selecting appropriate critical values $μ_{1}^{*}, \dots, μ_{q}^{*}$ of the parameters. The required continuity properties of the critical parameter functions will rely upon zero number arguments applied to the difference of two solutions.

In view of the proof, we need two propositions. The first one gives the building blocks of our multibump construction, namely the individual bumps and the linking functions between two bumps (cf. Fig. 5).

Proposition 9.2.

(i) There exist $0 < a_{1} < a_{2} < \frac{1}{4}$ , $c_{2} > c_{1} > 0$ , $K_{1} = K_{1} (p) > 0$ , $K > \tilde{K} (a_{1})$ (cf. Proposition 3.1 (i)) and, for all $ε \in (0, \frac{1}{2})$ , there exists a nonnegative function $ψ_{ε} \in C^{2} (R)$ , symmetric with respect to $x = ε$ , such that $\begin{array}{l} (9.2) & Supp (ψ_{ε}) = [(1 - a_{2}) ε, (1 + a_{2}) ε], \\ (9.3) & ψ_{ε} > K x^{α} on [(1 - a_{1}) ε, (1 + a_{1}) ε], \\ (9.4) & ‖ ψ_{ε} ‖_{\infty} ⩽ K_{1} ε^{α}, \\ (9.5) & ψ_{ε}^{'} > 0 on ((1 - a_{2}) ε, ε), ψ_{ε}^{″} ⩾ 0 on [0, 1] ∖ ((1 - a_{1}) ε, (1 + a_{1}) ε), \\ (9.6) & ψ_{ε} - U^{*} has exactly one zero in [(1 - a_{2}) ε, ε) and one zero in (ε, (1 + a_{2}) ε], \\ (9.7) & for any λ \in [0, 1], λ ψ_{ε} - U^{*} has at most 2 zeros in (0, 1] . \end{array}$ Moreover, with $c_{2} > c_{1} > 0$ given by Proposition 3.1 (ii), there exists $η = η (p) \in (0, 1)$ such that for any $τ \in [1 - η, 1]$ , $\begin{array}{l} (9.8) & u (τ ψ_{ε}; 0, t) > 0 for all t \in (c_{1} ε^{2}, c_{2} ε^{2}) . \end{array}$

(ii) Let $ε \in (0, 1)$ and $\bar{ε} \in (0, ε / 2]$ . Then $(1 + a_{2}) \bar{ε} < (1 - a_{2}) ε$ (hence the supports of $ψ_{\bar{ε}}$ and $ψ_{ε}$ are disjoint) and there exists a function $h = h_{\bar{ε}, ε} \in C^{2} ([0, 1])$ such that $\begin{array}{l} (9.9) & h ⩾ \hat{ψ} : = ψ_{\bar{ε}} + ψ_{ε}, h_{| [0, (1 + a_{1}) \bar{ε}]} = ψ_{\bar{ε}}, h_{| [(1 - a_{1}) ε, 1]} = ψ_{ε}, \\ (9.10) & h > K x^{α} and {(h - K x^{α})}^{″} ⩾ 0 in [(1 + a_{1}) \bar{ε}, (1 - a_{1}) ε], \\ (9.11) & ‖ h ‖_{\infty} ⩽ K_{1} ε^{α} . \end{array}$ Moreover, $\begin{array}{l} (9.12) & for any λ \in [0, 1], (λ h + (1 - λ) \hat{ψ}) - U^{*} has at most 2 zeros in [\bar{ε}, ε] . \end{array}$

Fig. 5.

The functions in Proposition 9.2 (the function h is the dotted curve).

In order not to interrupt the main flow of ideas, the proof of Proposition 9.2, which is somewhat technical, is postponed to the appendix.

In the second proposition, we prepare the preliminary sequence of bumps $ϕ_{1}, \dots, ϕ_{m}$ . This sequence is similar to that in the proof of Theorem 1.1, but needs more precise choices of parameters. Also, we introduce the corresponding linking functions $h_{i}$ , as well as majorizing functions $H_{i}$ that will be used in the deformation step (cf. Fig. 6).

Fig. 6.

The bumps and linking functions in Proposition 9.3 (for $m = 3$ ).

Proposition 9.3.

Let the constants $L, K_{1}, c_{1}, c_{2}$ and the functions $γ_{0}, ψ_{ε}, h_{\bar{ε}, ε}$ be defined in Propositions 3.2 and 9.2 , and set $c_{0} = (c_{1} + c_{2}) / 2$ . For each $m ⩾ 2$ , there exist numbers $0 < ε_{1} < \dots < ε_{m} < ε_{m + 1} < \frac{1}{4}$ , $γ_{1}, \dots, γ_{m + 1} > 0$ and functions $ϕ_{1}, \dots, ϕ_{m}$ , $h_{1}, \dots, h_{m}$ , $H_{1}, \dots, H_{m + 1} \in C^{2} ([0, 1])$ , with the following properties: $\begin{array}{l} (9.13) & ϕ_{i} = ψ_{ε_{i}}, i \in {1, \dots, m}, \\ (9.14) & h_{i} = h_{ε_{i}, ε_{i + 1}}, i \in {1, \dots, m - 1}, \\ (9.15) & h_{m} = ϕ_{m}, \\ (9.16) & H_{i} = max (h_{i}, \dots, h_{m}), i \in {1, \dots, m}, \\ (9.17) & H_{m + 1} = 0, \\ (9.18) & ‖ H_{1} ‖_{\infty} < 2^{- α} c_{p} and Supp (H_{1}) \subset (0, \frac{1}{2}), \\ (9.19) & γ_{i} = min {\frac{1}{4} γ_{0} (\frac{1}{2} ε_{i}, ‖ H_{i} ‖_{C^{2} ([0, 1])}), \frac{C_{1}}{2 L} ε_{i}^{2}}, i \in {1, \dots, m + 1}, \\ (9.20) & ε_{i - 1} = min {\frac{1}{8} ε_{i}, {(K_{1}^{- 1} γ_{i})}^{1 / α}, \frac{1}{2} {(c_{2}^{- 1} L γ_{i})}^{1 / 2}}, i \in {2, \dots, m + 1} . \end{array}$ Moreover, the functions $ϕ_{i}$ have disjoint supports and $\begin{array}{l} (9.21) & h_{i} ⩾ ϕ_{i} + ϕ_{i + 1}, i \in {1, \dots, m - 1} . \end{array}$ Finally, the above remains valid for $m = 1$ , omitting properties ( 9.14 ) and ( 9.21 ).

Proof.

In view of the application of Proposition 3.2, bumps will be sequentially added by moving toward the boundary $x = 0$ . Therefore, we proceed by backward induction, starting from $\begin{array}{l} (9.22) & ε_{m + 1} \in (0, min {\frac{1}{4}, \frac{1}{2} {(c_{p} / K_{1})}^{1 / α}}), H_{m + 1} = 0 . \end{array}$ Take $i \in {2, \dots, m + 1}$ , and assume that $0 < ε_{i} < \dots < ε_{m + 1}$ , $H_{i}, \dots, H_{m + 1} \in C^{2} ([0, 1])$ have already been chosen. If $i ⩽ m$ , assume in addition that $ϕ_{i}, \dots, ϕ_{m}$ , $h_{i}, \dots, h_{m} \in C^{2} ([0, 1])$ have already been chosen and that they satisfy $\begin{array}{l} (9.23) & H_{i} = ϕ_{i} in [0, (1 + a_{1}) ε_{i}], H_{i} ⩾ ϕ_{i} in [0, 1], \end{array}$ where $a_{1}$ is given by Proposition 9.2. We let $\begin{array}{l} γ_{i} = min {\frac{1}{4} γ_{0} (\frac{1}{2} ε_{i}, ‖ H_{i} ‖_{C^{2} ([0, 1])}), \frac{C_{1}}{2 L} ε_{i}^{2}} \end{array}$ where the function $γ_{0}$ is given by Proposition 3.2, omitting the dependence in $p, Ω$ for conciseness, and we next choose $\begin{array}{l} ε_{i - 1} = min {\frac{1}{8} ε_{i}, {(K_{1}^{- 1} γ_{i})}^{1 / α}, \frac{1}{2} {(c_{2}^{- 1} L γ_{i})}^{1 / 2}} . \end{array}$ With the notation of Proposition 9.2, we then set $\begin{array}{l} ϕ_{i - 1} = ψ_{ε_{i - 1}}, \\ h_{i - 1} = \{\begin{matrix} ϕ_{m}, & if i = m + 1, \\ h_{ε_{i - 1}, ε_{i}}, & if i ⩽ m, \end{matrix} \end{array}$ and $\begin{array}{l} (9.24) & H_{i - 1} = max (H_{i}, h_{i - 1}) . \end{array}$

Observe that, by (9.9) and (9.23), we have $H_{i - 1} = H_{i}$ on $[ε_{i}, 1]$ , $H_{i - 1} = h_{i - 1}$ on $[0, ε_{i})$ , and $H_{i - 1} = ϕ_{i}$ on $[(1 - a_{1}) ε_{i}, (1 + a_{1}) ε_{i}]$ . This in particular guarantees that $H_{i - 1} \in C^{2} ([0, 1])$ . Using (9.9) again, we also have $H_{i - 1} = ϕ_{i - 1}$ in $[0, (1 + a_{1}) ε_{i - 1}]$ and, in view of (9.24), $H_{i - 1} ⩾ ϕ_{i - 1}$ in $[0, 1]$ . Therefore, the new function $H_{i - 1}$ satisfies (9.23) with i replaced by $i - 1$ and we can carry out the iteration from $i = m + 1$ down to $i = 2$ .

Finally, we define $γ_{1} = min {\frac{1}{4} γ_{0} (\frac{1}{2} ε_{1}, ‖ H_{1} ‖_{C^{2} ([0, 1])}), \frac{C_{1}}{2 L} ε_{1}^{2}}$ . It is then easy to check that properties (9.13)–(9.20) are true, whereas the disjointness of the supports of the $ϕ_{i}$ and (9.21) respectively follow from (9.2) and (9.9). □

Proof of Theorem 1.4’.

Step 1. Preliminary multibump initial data. Our starting point is a multibump initial data, defined by the sum $\begin{array}{l} \hat{ϕ} : = \sum_{i = 1}^{m} ϕ_{i}, \end{array}$ where m will be determined herefater and the compactly supported space bumps $ϕ_{1}, \dots, ϕ_{m}$ are given by Proposition 9.3 (recall that they have disjoint supports). Suitable deformations will then be applied to $\hat{ϕ}$ . In the rest of the proof, we shall use the various constants and functions defined in Proposition 9.3.

Let us compute the number m of space bumps, and the deformation type of each bump, in terms of the given sequence $({\bar{σ}}_{i})$ (recall that we are proving Theorem 1.4 under the equivalent form of Theorem 1.4’). We first inductively define integers ${(κ_{i})}_{1 ⩽ i ⩽ d + 1}$ by setting $\begin{array}{l} (9.25) & κ_{1} = 1, κ_{i + 1} = κ_{i} + max (1, {\bar{σ}}_{i}) for i = 1, \dots, d . \end{array}$ We then set $m : = κ_{d + 1} - 1$ and define ${(ξ_{j})}_{1 ⩽ j ⩽ m}$ by: $\begin{array}{l} (9.26) & \{\begin{matrix} ξ_{κ_{i}} = {\bar{σ}}_{i} & if {\bar{σ}}_{i} ⩽ 1, \\ ξ_{κ_{i}} = \dots = ξ_{κ_{i} + {\bar{σ}}_{i} - 2} = 2, ξ_{κ_{i} + {\bar{σ}}_{i} - 1} = 1, & if {\bar{σ}}_{i} ⩾ 2 . \end{matrix} \end{array}$

The numbers $ξ_{i}$ will play the role of deformation indices. Namely, for all $i \in {1, \dots, m}$ , if $\begin{array}{l} ξ_{i} = \{\begin{matrix} 0, & then the amplitude of ϕ_{i} will be reduced, \\ 1, & then ϕ_{i} will be left unchanged, \\ 2, & then ϕ_{i} will be linked with its right neighbor \end{matrix} \end{array}$ (note that $ξ_{m} \neq 2$ by (9.26)). As indicated in the outline of proof, these operations are respectively aimed at producing GBU without LBC, LBC, and bouncing. For the example in Fig. 4 (where we had $d = 3$ , ${\bar{σ}}_{1} = 1$ , ${\bar{σ}}_{2} = 3$ and ${\bar{σ}}_{3} = 0$ ), we need $m = 5$ bumps. These bumps and their planned deformations, as well as the indices $κ_{i}$ , $ξ_{i}$ , are depicted in Fig. 7.

Fig. 7.

The bumps $ϕ_{i}$ and indices $κ_{i}$ , $ξ_{i}$ corresponding to the example in Fig. 4. The planned deformations are represented by the dotted curves.

Step 2. Deformation of initial data and their intersection properties. Denote by $1 ⩽ i_{1} < \dots < i_{q} ⩽ m$ the indices of the bumps to be deformed, i.e. the elements of ${1, \dots, m}$ such that $ξ_{i} \neq 1$ , and define the parameter space $J : = {[0, 1]}^{q}$ . In the rest of the proof we assume $q ⩾ 2$ . The much easier case $q = 1$ can be treated by obvious modifications. Also, for any $μ \in J$ and $j \in {1, \dots, q}$ , we shall denote ${\tilde{μ}}_{j} = (μ_{j}, \dots, μ_{q})$ , so that we can write $μ = (μ_{1}, \dots, μ_{j - 1}, {\tilde{μ}}_{j})$ for $j \in {2, \dots, q}$ .

We construct a q-parameter deformation of $\hat{ϕ}$ , denoted by $Φ_{μ}$ by setting $\begin{array}{l} (9.27) & Φ_{μ} = \hat{ϕ} + \sum_{\begin{array}{c} 1 ⩽ ℓ ⩽ q \\ ξ_{i_{ℓ}} = 0 \end{array}} (μ_{ℓ} - 1) ϕ_{i_{ℓ}} + \sum_{\begin{array}{c} 1 ⩽ ℓ ⩽ q \\ ξ_{i_{ℓ}} = 2 \end{array}} μ_{ℓ} {\tilde{ϕ}}_{i_{ℓ}}, μ = (μ_{1}, \dots, μ_{q}) \in J, \end{array}$ where ${\tilde{ϕ}}_{i_{ℓ}} : = h_{i_{ℓ}} - ϕ_{i_{ℓ}} - ϕ_{i_{ℓ} + 1} ⩾ 0$ (owing to (9.21)). Observe that the supports of the deformations satisfy $\begin{array}{l} (9.28) & \{\begin{matrix} Supp (ϕ_{i_{ℓ}}) \subset I_{ℓ}^{1} : = [(1 - a_{2}) ε_{i_{ℓ}}, (1 + a_{2}) ε_{i_{ℓ}}] & if ξ_{i_{ℓ}} = 0, \\ Supp ({\tilde{ϕ}}_{i_{ℓ}}) \subset I_{ℓ}^{2} : = [(1 + a_{1}) ε_{i_{ℓ}}, (1 - a_{1}) ε_{i_{ℓ} + 1}] & if ξ_{i_{ℓ}} = 2, \end{matrix} \end{array}$ by (9.2), (9.9), (9.13), (9.14), and all these intervals are disjoint (since for all $i \in {2, \dots, m}$ , $ξ_{i} = 0 \Rightarrow ξ_{i - 1} \neq 2$ , in view of (9.26)).

Setting $\begin{array}{l} {\hat{ε}}_{i} = \frac{3}{4} ε_{i}, 1 ⩽ i ⩽ m + 1, \end{array}$ and recalling that $ε_{i - 1} ⩽ \frac{1}{8} ε_{i}$ by (9.20), we have $\begin{array}{l} {\hat{ε}}_{1} < ε_{1} < {\hat{ε}}_{2} < \dots < {\hat{ε}}_{m} < ε_{m} < {\hat{ε}}_{m + 1} < ε_{m + 1} . \end{array}$ We note that for each $i \in {1, \dots, m}$ , $Φ_{μ}$ satisfies $\begin{array}{l} (9.29) & Φ_{μ} = \{\begin{matrix} \hat{ϕ} on [{\hat{ε}}_{i}, {\hat{ε}}_{i + 1}] & if ξ_{i} = 1 \\ μ_{ℓ} \hat{ϕ} on [{\hat{ε}}_{i}, {\hat{ε}}_{i + 1}] & if ξ_{i} = 0, i = i_{ℓ}, ℓ \in {1, \dots, q} \\ μ_{ℓ} h_{i} + (1 - μ_{ℓ}) \hat{ϕ} on [ε_{i}, ε_{i + 1}] & if ξ_{i} = 2, i = i_{ℓ}, ℓ \in {1, \dots, q} . \end{matrix} \end{array}$ For simplicity, we shall denote the corresponding solutions by $\begin{array}{l} u (μ; x, t) = u (Φ_{μ}; x, t), μ \in J . \end{array}$ Owing to (9.18), we also have $\begin{array}{l} (9.30) & ‖ Φ_{μ} ‖_{\infty} < c_{p}, μ \in J, \\ (9.31) & Supp (Φ_{μ}) \subset (0, \frac{1}{2}), μ \in J, \end{array}$ and there exists $a > 0$ such that $\begin{array}{l} (9.32) & Φ_{μ} (x) ⩽ U_{a} (1 - x), 0 ⩽ x ⩽ 1, μ \in J, \end{array}$ where the regular steady state $U_{a}$ is defined in (6.3). It follows from (9.32) and Proposition 2.2 that $u (μ; x, t) ⩽ U_{a} (1 - x)$ for all $(x, t) \in [0, 1] \times [0, \infty)$ . As a consequence of Lemma 5.1(ii) (applied to the reflected solution $\tilde{u} (x, t) = u (1 - x, t)$ ), we have $\begin{array}{l} (9.33) & u (μ; \cdot, \cdot) \in C^{2, 1} ([\frac{1}{2}, 1] \times (0, \infty)) \end{array}$ and $\begin{array}{l} (9.34) & u (μ; 1, t) = 0, t ⩾ 0, μ \in J . \end{array}$

For all $μ \in J$ and $i \in {1, \dots, m}$ , as a direct consequence of (9.29) and Proposition 9.2, the initial data $Φ_{μ}$ enjoy the following intersection properties (these properties, for the example from Fig. 4, are illustrated in Fig. 8): $\begin{array}{l} (9.35) & ξ_{i} = 2 ⟹ z_{| [ε_{i}, ε_{i + 1}]} (Φ_{μ} - U^{*}) ⩽ 2 with Φ_{μ} - U^{*} > 0 in {ε_{i}, ε_{i + 1}}, \\ (9.36) & ξ_{i} = 0 ⟹ z_{| [{\hat{ε}}_{i}, {\hat{ε}}_{i + 1}]} (Φ_{μ} - U^{*}) ⩽ 2 with Φ_{μ} - U^{*} < 0 in {{\hat{ε}}_{i}, {\hat{ε}}_{i + 1}}, \end{array}$ and we also have $\begin{array}{l} (9.37) & \begin{aligned} ξ_{i} = 1 ⟹ z_{| [ε_{i}, {\hat{ε}}_{i + 1}]} (Φ_{μ} - U^{*}) = 1 \\ with (Φ_{μ} - U^{*}) (ε_{i}) > 0 > (Φ_{μ} - U^{*}) ({\hat{ε}}_{i + 1}) \end{aligned} \end{array}$ and $\begin{array}{l} (9.38) & \begin{aligned} (ξ_{i} \neq 0, with ξ_{i - 1} \neq 2 if i ⩾ 2) ⟹ z_{| [{\hat{ε}}_{i}, ε_{i}]} (Φ_{μ} - U^{*}) = 1 \\ with (Φ_{μ} - U^{*}) ({\hat{ε}}_{i}) < 0 < (Φ_{μ} - U^{*}) (ε_{i}) . \end{aligned} \end{array}$ In particular, by an easy induction argument, it follows that $\begin{array}{l} (9.39) & z_{| [0, 1]} (Φ_{μ} - U^{*}) ⩽ 2 m . \end{array}$

Fig. 8.

Sign changes of $ϕ_{μ} - U^{*}$ (for $m = 5$ , $q = 3$ with $i_{1} = 2$ , $i_{2} = 3$ , $i_{3} = 5$ and $ξ_{2} = ξ_{3} = 2$ , $ξ_{5} = 0$ ).

On the other hand, we observe that, for all $j \in {1, \dots, q - 1}$ and $λ, μ \in J$ , we have $\begin{array}{l} (9.40) & {\tilde{λ}}_{j + 1} = {\tilde{μ}}_{j + 1} ⟹ z_{| [0, 1]} (Φ_{μ} - Φ_{λ}) ⩽ j - 1 . \end{array}$ Indeed, in view of $ϕ_{i}, {\tilde{ϕ}}_{i} ⩾ 0$ and (9.27), for each $ℓ \in {1, \dots, j}$ , the function $Φ_{μ} - Φ_{λ}$ does not change sign in the interval $I_{ℓ}^{1}$ if $ξ_{i_{ℓ}} = 0$ and $I_{ℓ}^{2}$ if $ξ_{i_{ℓ}} = 2$ (cf. (9.28)), whereas $Φ_{μ} - Φ_{λ} \equiv 0$ outside of these j intervals.

Step 3. LBC and regularity properties of the deformed solutions.

Define the times $\begin{array}{l} s_{i} = c_{0} ε_{i}^{2}, s_{i}^{-} = c_{1} ε_{i}^{2}, s_{i}^{+} = c_{2} ε_{i}^{2}, i \in {1, \dots, m}, \\ {\hat{s}}_{i} = \frac{3}{2} L γ_{i}, {\hat{s}}_{i}^{-} = L γ_{i}, {\hat{s}}_{i}^{+} = 2 L γ_{i}, i \in {1, \dots, m + 1} . \end{array}$ By (9.19), (9.20), we note that (cf. Fig. 9) $\begin{array}{l} \dots < {\hat{s}}_{i}^{-} < {\hat{s}}_{i} < {\hat{s}}_{i}^{+} < s_{i}^{-} < s_{i} < s_{i}^{+} < {\hat{s}}_{i + 1}^{-} < {\hat{s}}_{i + 1} < {\hat{s}}_{i + 1}^{+} < \dots \end{array}$

Fig. 9.

Time behavior of the deformed solutions for $i = i_{ℓ}$ with $ξ_{i_{ℓ}} = 0$ .

We claim that for all $μ \in J$ , we have: $\begin{array}{l} If i \in {1, m + 1} or if i \in {2, \dots, m} and ξ_{i - 1} \neq 2, then \\ (9.41) & u (μ; \cdot, \cdot) is classical on [{\hat{s}}_{i}^{-}, {\hat{s}}_{i}^{+}], \\ If i \in {1, \dots, m} and ξ_{i} \neq 0, then \\ (9.42) & u (μ; 0, s) > 0 for all s \in [s_{i}^{-}, s_{i}^{+}], \\ If ℓ \in {1, \dots, q} and ξ_{i_{ℓ}} = 0, then \\ (9.43) & \{\begin{matrix} u (μ; \cdot, \cdot) is classical on [{\hat{s}}_{i_{ℓ}}, {\hat{s}}_{i_{ℓ} + 1}], & if μ_{ℓ} = 0 \\ u (μ; 0, s) > 0 for all s \in [s_{i_{ℓ}}^{-}, s_{i_{ℓ}}^{+}], & if μ_{ℓ} is close to 1, \end{matrix} \\ If ℓ \in {1, \dots, q} and ξ_{i_{ℓ}} = 2 (hence i_{ℓ} ⩽ m - 1), then \\ (9.44) & \{\begin{matrix} u (μ; \cdot, \cdot) is classical on [{\hat{s}}_{i_{ℓ} + 1}^{-}, {\hat{s}}_{i_{ℓ} + 1}^{+}], & if μ_{ℓ} = 0 \\ u (μ; 0, s) > 0 for all s \in [s_{i_{ℓ}}, s_{i_{ℓ} + 1}], & if μ_{ℓ} is close to 1 \end{matrix} \end{array}$ (these behaviors are illustrated in Fig. 9–10).

Fig. 10.

Time behavior of the deformed solutions for $i = i_{ℓ}$ with $ξ_{i_{ℓ}} = 2$ .

Let us first check the regularity properties in (9.41) and (9.44). Let $i \in {2, \dots, m + 1}$ , and in case $ξ_{i - 1} = 2$ assume $i = i_{ℓ} + 1$ with $μ_{ℓ} = 0$ . By (9.2), (9.9), (9.14) and (9.16), we have $\begin{array}{l} Φ_{μ} ⩽ H_{i} on [\frac{1}{2} ε_{i}, 1] . \end{array}$ On the other hand, since $(1 + a_{2}) ε_{i - 1} < \frac{1}{2} ε_{i}$ by (9.20), it follows from (9.2), (9.4), (9.11) that $\begin{array}{l} Φ_{μ} ⩽ K_{1} ε_{i - 1}^{α} ⩽ γ_{i} on [0, \frac{1}{2} ε_{i}] . \end{array}$ Also, if $i = 1$ , then $\begin{array}{l} Φ_{μ} = 0 on [0, \frac{1}{2} ε_{1}] . \end{array}$ Therefore, recalling (9.19), (9.31), Proposition 3.2 guarantees that the solution $u (μ; \cdot, \cdot)$ is classical on the time interval $[L γ_{i}, L γ_{0} (\frac{1}{2} ε_{i}, ‖ H_{i} ‖_{C^{2} ([0, 1])})] \supset [{\hat{s}}_{i}^{-}, {\hat{s}}_{i}^{+}]$ . This proves (9.41) and the first part of (9.44).

Next assume $i = i_{ℓ}$ with $ξ_{i_{ℓ}} = 0$ and $μ_{ℓ} = 0$ . By (9.2), (9.9), (9.14) and (9.16), we have $\begin{array}{l} Φ_{μ} ⩽ H_{i + 1} on [\frac{1}{2} ε_{i + 1}, 1] . \end{array}$ On the other hand, in case $i ⩾ 2$ , since $(1 + a_{2}) ε_{i - 1} < 2 ε_{i - 1}$ and $\frac{1}{2} ε_{i + 1} < (1 - a_{2}) ε_{i + 1}$ , we have $\begin{array}{l} (9.45) & Φ_{μ} = 0 on [2 ε_{i - 1}, \frac{1}{2} ε_{i + 1}] and Φ_{μ} ⩽ K_{1} ε_{i - 1}^{α} ⩽ γ_{i} on [0, 2 ε_{i - 1}), \end{array}$ Also (9.45) remains true for $i = 1$ with $ε_{0} : = 0$ (omitting the second condition). In both cases we thus have $\begin{array}{l} Φ_{μ} ⩽ γ_{i} on [0, \frac{1}{2} ε_{i + 1}] . \end{array}$ Therefore, in view of (9.19), (9.31), Proposition 3.2 guarantees that $u (μ; \cdot, \cdot)$ is classical on the time interval $[L γ_{i}, L γ_{0} (\frac{1}{2} ε_{i + 1}, ‖ H_{i + 1} ‖_{C^{2} ([0, 1])})]$ . Since, by (9.19), this interval contains $[{\hat{s}}_{i}, {\hat{s}}_{i + 1}]$ , the first part of (9.43) follows.

As for the LBC properties, (9.42) and the second part of (9.43) follow from (9.8). To check the second part of (9.44), note that if $i = i_{ℓ}$ with $ξ_{i_{ℓ}} = 2$ and $μ_{ℓ}$ is close to 1, then (9.3) and (9.10) guarantee that $\begin{array}{l} Φ_{μ} (x) ⩾ K x^{α} on ((1 - a_{1}) x, (1 + a_{1}) x) for all x \in [ε_{i - 1}, ε_{i}] . \end{array}$ Since $K > \tilde{K} (a_{1})$ , the conclusion follows from Proposition 3.1(ii).

Step 4. Definition of the critical values.

For any $σ \in {0, 2}$ , any closed subinterval J of $[0, \infty)$ and continuous function w on ${0} \times J$ , we set $\begin{array}{l} E (w, J, σ) = \{\begin{matrix} {max}_{J} w & if σ = 0 \\ {min}_{J} w & if σ = 2 \end{matrix} \end{array}$ and we define the intervals $\begin{array}{l} J_{ℓ} = \{\begin{matrix} [{\hat{s}}_{i_{ℓ}}, {\hat{s}}_{i_{ℓ} + 1}] & if ξ_{ℓ} = 0, \\ [s_{i_{ℓ}}, s_{i_{ℓ} + 1}] & if ξ_{ℓ} = 2, \end{matrix} ℓ \in {1, \dots, q} \end{array}$ (the intervals $J_{ℓ}$ are represented by horizontal dotted lines in Fig. 9–10).

We shall iteratively define q critical parameter functions $\begin{array}{l} μ_{1}^{*} ({\tilde{μ}}_{2}), \dots, μ_{q - 1}^{*} ({\tilde{μ}}_{q}), μ_{q}^{*} \end{array}$ and $q + 1$ auxiliary solutions $\begin{array}{l} u_{0}^{*} ({\tilde{μ}}_{1}, x, t), u_{1}^{*} ({\tilde{μ}}_{2}, x, t), \dots, u_{q - 1}^{*} ({\tilde{μ}}_{q}; x, t), u_{q}^{*} (x, t) \end{array}$ as follows. Set $\begin{array}{l} (9.46) & u_{0}^{*} ({\tilde{μ}}_{1}, x, t) = u ({\tilde{μ}}_{1}, x, t) . \end{array}$ Take $j \in {1, \dots, q}$ and assume we have already defined $u_{j - 1}^{*} ({\tilde{μ}}_{j}; x, t)$ . Then, for any ${\tilde{μ}}_{j + 1} \in {[0, 1]}^{q - j}$ , we consider the set $\begin{array}{l} E_{j} ({\tilde{μ}}_{j + 1}) : = {μ_{j} \in [0, 1]; E (u_{j - 1}^{*} (μ_{j}, {\tilde{μ}}_{j + 1}; 0, \cdot), J_{j}, ξ_{i_{j}}) > 0} \end{array}$ (understood as $E_{q} : = {μ_{q} \in [0, 1]; E (u_{q - 1}^{*} (μ_{q}; 0, \cdot), J_{q}, ξ_{i_{q}}) > 0}$ if $j = q$ ). We note that $E_{j} ({\tilde{μ}}_{j + 1}) \neq \emptyset$ since $1 \in E_{j} ({\tilde{μ}}_{j + 1})$ by the second parts of (9.43)–(9.44). We then define $\begin{array}{l} μ_{j}^{*} ({\tilde{μ}}_{j + 1}) = inf E_{j} ({\tilde{μ}}_{j + 1}) \end{array}$ and $\begin{array}{l} (9.47) & u_{j}^{*} ({\tilde{μ}}_{j + 1}; x, t) = u_{j - 1}^{*} (μ_{j}^{*} ({\tilde{μ}}_{j + 1}), {\tilde{μ}}_{j + 1}; x, t) . \end{array}$ (respectively understood as $μ_{q}^{*} = inf E_{q}$ and $u_{q}^{*} (x, t) = u_{q - 1}^{*} (μ_{q}^{*}; x, t)$ if $j = q$ ). Similarly, in what follows we shall make the convention that the variable ${\tilde{μ}}_{q + 1}$ is empty.

For further use, we immediately note by induction that each of the functions $u_{j}^{*} ({\tilde{μ}}_{j + 1}; \cdot, \cdot)$ is the solution of (1.1) with initial data $Φ_{Θ}$ for some $Θ \in J$ , namely: $\begin{array}{l} (9.48) & \begin{aligned} for any j \in {0, \dots, q} and {\tilde{μ}}_{j + 1} \in {[0, 1]}^{q - j}, there exists \\ (λ_{1}, \dots, λ_{j}) \in {[0, 1]}^{j} such that u_{j}^{*} ({\tilde{μ}}_{j + 1}; \cdot, \cdot) = u (λ_{1}, \dots, λ_{j}, {\tilde{μ}}_{j + 1}; \cdot, \cdot) . \end{aligned} \end{array}$

Step 5. Properties of the critical parameter functions $μ_{j}^{*} ({\tilde{μ}}_{j + 1})$ .

Let $Q = \overline{Ω} \times (0, \infty)$ . Let D be the constant given by Lemma 5.1(ii) (denoted there by K), which can be chosen uniformly with respect to $μ \in J$ , owing to ${sup}_{μ \in J} ‖ ϕ_{μ} ‖_{C^{2} ([0, 1])} < \infty$ . Set $V_{D} (x) : = U^{*} (x) - D x^{2}$ .

We shall prove by joint induction that, for all $j \in {1, \dots, q - 1}$ , $\begin{array}{l} for all {\tilde{μ}}_{j + 1} \in {[0, 1]}^{q - j} and ℓ \in {1, \dots, j}, \\ there exists T_{ℓ} = T_{ℓ} ({\tilde{μ}}_{j + 1}) \in J_{ℓ} s.t. \\ (9.49) & \{\begin{matrix} u_{j}^{*} ({\tilde{μ}}_{j + 1}; \cdot, T_{ℓ}) ⩾ V_{D} in (0, 1) and u_{j}^{*} ({\tilde{μ}}_{j + 1}; 0, \cdot) = 0 in J_{ℓ} & if ξ_{i_{ℓ}} = 0 \\ u_{j}^{*} ({\tilde{μ}}_{j + 1}; \cdot, \cdot) ⩾ V_{D} in (0, 1) \times J_{ℓ} and u_{j}^{*} ({\tilde{μ}}_{j + 1}; 0, T_{ℓ}) = 0 & if ξ_{i_{ℓ}} = 2; \end{matrix} \\ (9.50) & the map {[0, 1]}^{q - j} ∋ {\tilde{μ}}_{j + 1} \mapsto μ_{j}^{*} ({\tilde{μ}}_{j + 1}) is continuous; \\ (9.51) & the map {[0, 1]}^{q - j} ∋ {\tilde{μ}}_{j + 1} \mapsto u_{j}^{*} ({\tilde{μ}}_{j + 1}, x, t) is continuous, \\ uniformly w.r.t. (x, t) \in Q, \end{array}$ and that $\begin{array}{l} for all ℓ \in {1, \dots, q}, there exists T_{ℓ} \in J_{ℓ} s.t. \\ (9.52) & \{\begin{matrix} u_{q}^{*} (\cdot, T_{ℓ}) ⩾ V_{D} in (0, 1) and u_{q}^{*} (0, \cdot) = 0 in J_{ℓ} & if ξ_{i_{ℓ}} = 0 \\ u_{q}^{*} (\cdot, \cdot) ⩾ V_{D} in (0, 1) \times J_{ℓ} and u_{q}^{*} (0, T_{ℓ}) = 0 & if ξ_{i_{ℓ}} = 2 \end{matrix} \end{array}$ (note that (9.52) corresponds to (9.49) with $j = q$ ).

To initialize the induction, we observe that the continuity property (9.51) is true for $j = 0$ (indeed, since $u_{0}^{*} ({\tilde{μ}}_{1}, x, t) = u (Φ_{μ}; x, t)$ , it follows from (2.3) and the fact that $Φ_{μ}$ depends continuously in $L^{\infty} (Ω)$ -norm on the parameters μ). This is the only part of the induction hypotheses (9.49)–(9.51) that will be used in the first induction step $j = 1$ . Thus take $j \in {1, \dots, q}$ and, if $j ⩾ 2$ , assume that (9.49)– ${(9.51)}_{j - 1}$ are true.3

Here and below this notation of course means that the formulae are understood with the index j replaced by the index in subscript (here $j - 1$ ).

First observe that, for all ${\tilde{μ}}_{j + 1} \in {[0, 1]}^{q - j}$ , if $ξ_{j} = 0$ (resp., $ξ_{j} = 2$ ) then we have $\begin{array}{l} μ_{j}^{*} ({\tilde{μ}}_{j + 1}) = inf {λ \in [0, 1]; max_{t \in J_{j}} u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; 0, t) > 0} (resp., min_{t \in J_{j}}) . \end{array}$ We note that $μ_{j}^{*} ({\tilde{μ}}_{j + 1}) \in [0, 1)$ due to (9.43)–(9.44). Therefore, there exist sequences $t_{k} \in J_{j}$ and $λ_{+}^{k} \to μ_{j}^{*} {({\tilde{μ}}_{j + 1})}^{+}$ as $k \to \infty$ such that, for $k ⩾ 1$ , $\begin{array}{l} (9.53) & \{\begin{matrix} u_{j - 1}^{*} (λ_{+}^{k}, {\tilde{μ}}_{j + 1}; 0, t_{k}) > 0 & if ξ_{i_{j}} = 0 \\ u_{j - 1}^{*} (λ_{+}^{k}, {\tilde{μ}}_{j + 1}; 0, \cdot) > 0 in J_{j} & if ξ_{i_{j}} = 2 . \end{matrix} \end{array}$ Also, if $μ_{j}^{*} ({\tilde{μ}}_{j + 1}) > 0$ , then there exist sequences ${\hat{t}}_{k} \in J_{j}$ and $λ_{-}^{k} \to μ_{j}^{*} {({\tilde{μ}}_{j + 1})}^{-}$ as $k \to \infty$ such that, for $k ⩾ 1$ , $\begin{array}{l} (9.54) & \{\begin{matrix} u_{j - 1}^{*} (λ_{-}^{k}, {\tilde{μ}}_{j + 1}; 0, \cdot) = 0 in J_{j} & if ξ_{i_{j}} = 0 \\ u_{j - 1}^{*} (λ_{-}^{k}, {\tilde{μ}}_{j + 1}; 0, {\hat{t}}_{k}) = 0 & if ξ_{i_{j}} = 2 . \end{matrix} \end{array}$

To prove ${(9.49)}_{j}$ , or (9.52) if $j = q$ , it suffices to check it in the case $ℓ = j$ (since for $ℓ \in {1, \dots, j - 1}$ it directly follows from (9.47) and our induction hypothesis). By Lemma 5.1(ii) and (9.53), we have $\begin{array}{l} (9.55) & \{\begin{matrix} u_{j - 1}^{*} (λ_{+}^{k}, {\tilde{μ}}_{j + 1}; \cdot, t_{k}) ⩾ V_{D} in (0, 1) & if ξ_{i_{j}} = 0 \\ u_{j - 1}^{*} (λ_{+}^{k}, {\tilde{μ}}_{j + 1}; \cdot, \cdot) ⩾ V_{D} in J_{j} \times (0, 1) & if ξ_{i_{j}} = 2 . \end{matrix} \end{array}$ Passing to a subsequence, there exists $T_{j} = T_{j} ({\tilde{μ}}_{j + 1}) \in J_{j}$ such that $t_{k} \to T_{j}$ , ${\hat{t}}_{k} \to T_{j}$ as $k \to \infty$ . Owing to the continuity property ${(9.51)}_{j - 1}$ , we may pass to the limit $k \to \infty$ in (9.55). In view of (9.47), this yields the inequalities in ${(9.49)}_{j}$ . If $μ_{j}^{*} ({\tilde{μ}}_{j + 1}) > 0$ , then we similarly get the equalities in ${(9.49)}_{j}$ by passing to the limit in (9.54) whereas, in case $μ_{j}^{*} ({\tilde{μ}}_{j + 1}) = 0$ , these equalities directly follow from (9.43)–(9.44).

We next assume $j ⩽ q - 1$ and prove the upper semicontinuity in ${(9.50)}_{j}$ . By (9.53) and the continuity property ${(9.51)}_{j - 1}$ , for each $k ⩾ 1$ , there exists $α_{k} > 0$ such that, for all $ζ \in {[0, 1]}^{q - j}$ with $| {\tilde{μ}}_{j + 1} - ζ | ⩽ α_{k}$ , we have $u_{j - 1}^{*} (λ_{+}^{k}, ζ; 0, t_{k}) > 0$ if $ξ_{j} = 0$ , and $u_{j - 1}^{*} (λ_{+}^{k}, ζ; 0, \cdot) > 0$ in $J_{j}$ if $ξ_{j} = 2$ . Consequently, $μ_{j}^{*} (ζ) ⩽ λ_{+}^{k}$ and, letting $k \to \infty$ , we thus get $\begin{array}{l} \underset{ζ \to {\tilde{μ}}_{j + 1}}{lim sup} μ_{j}^{*} (ζ) ⩽ μ_{j}^{*} ({\tilde{μ}}_{j + 1}) . \end{array}$

We now turn to the proof of the lower semicontinuity in ${(9.50)}_{j}$ , which is more delicate. Assume for contradiction that $\begin{array}{l} (9.56) & λ : = \underset{ζ \to {\tilde{μ}}_{j + 1}}{lim inf} μ_{j}^{*} (ζ) < μ_{j}^{*} ({\tilde{μ}}_{j + 1}) . \end{array}$ Arguing as above, there exist sequences $ζ^{k} \to {\tilde{μ}}_{j + 1}$ , ${\hat{λ}}_{\pm}^{k} \to λ$ as $k \to \infty$ and ${\hat{t}}_{k} \in J_{j}$ such that, for $k ⩾ 1$ , $\begin{array}{l} \{\begin{matrix} u_{j - 1}^{*} ({\hat{λ}}_{-}^{k}, ζ^{k}; 0, \cdot) = 0 in J_{j} and u_{j - 1}^{*} ({\hat{λ}}_{+}^{k}, ζ^{k}; 0, {\hat{t}}_{k}) ⩾ V_{D} in (0, 1) & if ξ_{i_{j}} = 0 \\ u_{j - 1}^{*} ({\hat{λ}}_{+}^{k}, ζ^{k}; 0, \cdot) ⩾ V_{D} in (0, 1) \times J_{j} and u_{j - 1}^{*} ({\hat{λ}}_{-}^{k}, ζ^{k}; 0, {\hat{t}}_{k}) = 0 & if ξ_{i_{j}} = 2 . \end{matrix} \end{array}$ Passing to a subsequence, there exists ${\hat{T}}_{j} \in J_{j}$ such that ${\hat{t}}_{k} \to {\hat{T}}_{j}$ as $k \to \infty$ . Using the continuity property ${(9.51)}_{j - 1}$ and passing to the limit $k \to \infty$ , we get $\begin{array}{l} (9.57) & \{\begin{matrix} u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; 0, \cdot) = 0 in J_{j} and u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; 0, {\hat{T}}_{j}) ⩾ V_{D} in (0, 1) & if ξ_{i_{j}} = 0 \\ u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; 0, \cdot) ⩾ V_{D} in (0, 1) \times J_{j} and u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; 0, {\hat{T}}_{j}) = 0 & if ξ_{i_{j}} = 2 . \end{matrix} \end{array}$ Denoting $μ = μ_{j}^{*} ({\tilde{μ}}_{j + 1})$ , we set $\begin{array}{l} (9.58) & \begin{aligned} v (x, t) : = u_{j - 1}^{*} (μ, {\tilde{μ}}_{j + 1}; x, t) = u_{j}^{*} ({\tilde{μ}}_{j + 1}; x, t), \\ w (x, t) : = u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; x, t) . \end{aligned} \end{array}$ We then use ${(9.49)}_{j}$ applied to $u_{j}^{*} ({\tilde{μ}}_{j + 1}; x, t)$ , (9.57) and, in case $j ⩾ 2$ , ${(9.49)}_{j - 1}$ applied to $u_{j - 1}^{*} (λ, {\tilde{μ}}_{j + 1}; x, t)$ . It follows that, for all $ℓ \in {1, \dots, j}$ , $\begin{array}{l} (9.59) & \{\begin{matrix} v (0, \cdot) = w (0, \cdot) = 0 in J_{ℓ} & if ξ_{i_{ℓ}} = 0 \\ v_{x} (0, \cdot) = w_{x} (0, \cdot) = \infty in J_{ℓ} & if ξ_{i_{ℓ}} = 2 \end{matrix} \end{array}$ and there exist $T_{ℓ}, {\hat{T}}_{ℓ} \in J_{ℓ}$ such that $\begin{array}{l} (9.60) & \{\begin{matrix} v_{x} (0, T_{ℓ}) = w_{x} (0, {\hat{T}}_{ℓ}) = \infty & if ξ_{i_{ℓ}} = 0 \\ v (0, T_{ℓ}) = w (0, {\hat{T}}_{ℓ}) = 0 & if ξ_{i_{ℓ}} = 2 . \end{matrix} \end{array}$

To reach a contradiction, the idea is now to apply the natural scaling to w and to examine the dropping properties of their number of intersections with the solution v. Namely, for $β \in (0, 1)$ , set $\begin{array}{l} (9.61) & w_{β} (x, t) = β^{α} w (β^{- 1} x, β^{- 2} t), (x, t) \in [0, β] \times [0, \infty) \end{array}$ and note that $w_{β}$ is the viscosity solution of problem (1.1) with $Ω = (0, β)$ and with initial data $β^{α} w (β^{- 1} x, 0)$ . Moreover, by (9.34), we have $\begin{array}{l} (9.62) & w_{β} (β, t) = 0, t ⩾ 0 . \end{array}$ For $β \in (0, 1]$ , we set $\begin{array}{l} N_{β} (t) : = z_{| [0, β]} (v (\cdot, t) - w_{β} (\cdot, t)), t > 0 . \end{array}$ By Proposition 6.2 and (9.62) we have $\begin{array}{l} (9.63) & for all β \in (0, 1), N_{β} (t) is finite and nonincreasing on [0, \infty) . \end{array}$ Also, by (9.40), (9.48) and (9.58), we have $\begin{array}{l} (9.64) & N_{1} (0) ⩽ j - 1 . \end{array}$

Let $τ_{0} : = (0, min (T_{1}^{*}, T_{2}^{*}))$ , where $T_{1}^{*}, T_{2}^{*}$ denote the classical existence times of $v, w$ . We claim that there exist $β_{0} \in (0, 1)$ and $τ \in (0, \frac{1}{2} τ_{0})$ such that, $\begin{array}{l} (9.65) & N_{β} (τ) ⩽ j - 1 for all β \in [β_{0}, 1) . \end{array}$ To show this, we first note that, by (9.27), (9.48), (9.58) and since $μ > λ$ due to (9.56), there exists $η \in (0, 1)$ such that $v (\cdot, 0) ⩾ w (\cdot, 0)$ in $[η, 1]$ and $v (η, 0) > w (η, 0)$ . By continuity, there exists $τ_{1} \in (0, τ_{0})$ such that $v (η, t) > w (η, t)$ for all $t \in [0, τ_{1}]$ , hence $\begin{array}{l} (9.66) & v > w in [η, 1) \times (0, τ_{1}] \end{array}$ by the strong maximum principle. Also, by standard zero number properties for classical solutions we may choose $τ \in (0, τ_{1})$ such that $\begin{array}{l} (9.67) & all zeros of v (\cdot, τ) - w (\cdot, τ) in [0, 1] are nondegenerate. \end{array}$ Next extending w by odd reflection at $x = 1$ , we obtain a function $\tilde{w} \in C^{1} ([0, 2] \times [0, τ_{0}))$ and, setting ${\tilde{w}}_{β} (x, t) = β^{α} \tilde{w} (β^{- 1} x, β^{- 2} t)$ , we see that ${\tilde{w}}_{β} (\cdot, τ) \to \tilde{w} (\cdot, τ)$ in $C^{1} ([0, 1])$ as $β \to 1^{-}$ . By (9.64) and (9.67), it follows that $\begin{array}{l} z ({\tilde{w}}_{β} (\cdot, τ) - v (\cdot, τ)) = N_{1} (τ) ⩽ N_{1} (0) ⩽ j - 1 \end{array}$ for β close to 1. Assuming also $β > η$ and using (9.66), along with the fact that $v > 0 > {\tilde{w}}_{β}$ for $x \in (β, 1)$ , we deduce (9.65).

Next, writing $J_{ℓ} = [t_{ℓ}^{-}, t_{ℓ}^{+}]$ , we claim the existence of $\bar{β} \in (β_{0}, 1)$ such that $\begin{array}{l} (9.68) & for all β \in (\bar{β}, 1), N_{β} (t_{ℓ}^{+}) ⩽ N_{β} (t_{ℓ}^{-}) - 1, ℓ = 1, \dots, j . \end{array}$ To prove the claim, assume for contradiction that there exist $ℓ \in {1, \dots, j}$ and a sequence $β_{k} \to 1$ with $β_{k} \in (β_{0}, 1)$ , such that $N_{β_{k}} (t_{ℓ}^{+}) ⩾ N_{β_{k}} (t_{ℓ}^{-})$ . By (9.63), for each k, we have $N_{β_{k}} (t) = N_{β_{k}} (t_{ℓ}^{-})$ on $J_{ℓ}$ , hence we may apply Lemma 7.5 to deduce the existence of $a_{k} \in (0, β_{k})$ such that $\begin{array}{l} (9.69) & v ⩽ w_{β_{k}} in (0, a_{k}] \times J_{ℓ} or v ⩾ w_{β_{k}} in (0, a_{k}] \times J_{ℓ} . \end{array}$ As a consequence of (9.41), (9.42), we have ${β_{k}}^{- 2} T_{ℓ}, {β_{k}}^{2} {\hat{T}}_{ℓ} \in J_{ℓ}$ for all $k ⩾ k_{0}$ large enough. It follows from (9.60) and (9.69) that $\begin{array}{l} (9.70) & \{\begin{matrix} w_{x} (0, {β_{k}}^{- 2} T_{ℓ}) = \infty or v_{x} (0, {β_{k}}^{2} {\hat{T}}_{ℓ}) = \infty, & if ξ_{i_{ℓ}} = 0 \\ v (0, {β_{k}}^{2} {\hat{T}}_{ℓ}) = 0 or w (0, {β_{k}}^{- 2} T_{ℓ}) = 0, & if ξ_{i_{ℓ}} = 2 . \end{matrix} \end{array}$ We thus deduce from (9.59) and (9.70) that, for all $k ⩾ k_{0}$ , ${β_{k}}^{2} {\hat{T}}_{ℓ} \in T_{v}$ or ${β_{k}}^{- 2} T_{ℓ} \in T_{w}$ . But since the transition sets $T_{v}$ and $T_{w}$ are finite by Theorem 1.3, this is impossible and claim (9.68) follows.

Finally, in view of (9.63) and since the intervals $J_{ℓ}$ have disjoint interiors, (9.68) contradicts (9.65). Therefore hypothesis (9.56) cannot be valid and we have proved ${(9.50)}_{j}$ .

As for property ${(9.51)}_{j}$ with $j ⩽ q - 1$ , it follows from ${(9.51)}_{j - 1}$ , ${(9.50)}_{j}$ and (9.47). We have thus proved (9.49)– ${(9.51)}_{j}$ for all $j \in {1, \dots, q - 1}$ and (9.52) by induction.

Step 6. Conclusion. We finally select the critical values of the parameters $μ : = Λ$ , where $Λ \in {[0, 1]}^{q}$ is defined by induction as follows: $\begin{array}{l} (9.71) & Λ_{q} = μ_{q}^{*} and Λ_{q - j} = μ_{q - j}^{*} (Λ_{q - j + 1}, \dots, Λ_{q}), j = 1, \dots, q - 1 . \end{array}$ Let us check that the solution $u (x, t) = u (Λ; x, t)$ , corresponding to the initial data $ϕ = Φ_{Λ}$ , has all the required properties.

We observe that, for all $j \in {0, \dots, q}$ , we have $\begin{array}{l} (9.72) & u (x, t) = u_{j}^{*} ({\tilde{Λ}}_{j + 1}; x, t), \end{array}$ where we recall that the notation $u_{q}^{*} ({\tilde{μ}}_{q + 1}; x, t)$ is understood as $u_{q}^{*} (x, t)$ (cf. after (9.47)). Indeed (9.72) follows by induction, noting that it is true for $j = 0$ by definition (cf. (9.46)) and that, if (9.72) is true at the level $j - 1$ for some $j \in {1, \dots, q}$ , then (9.47) and (9.71) give $\begin{array}{l} u_{j}^{*} ({\tilde{Λ}}_{j + 1}; x, t) = u_{j - 1}^{*} (μ_{j}^{*} ({\tilde{Λ}}_{j + 1}), {\tilde{Λ}}_{j + 1}; x, t) = u_{j - 1}^{*} ({\tilde{Λ}}_{j}; x, t) = u (x, t) . \end{array}$ Then applying (9.52) and (9.72) with $j = q$ , it follows that, for all $ℓ \in {1, \dots, q}$ , there exists $T_{ℓ} \in J_{ℓ}$ such that $\begin{array}{l} (9.73) & \{\begin{matrix} u (0, \cdot) = 0 in J_{ℓ} and u_{x} (0, T_{ℓ}) = \infty & if ξ_{i_{ℓ}} = 0 \\ u_{x} (0, \cdot) = \infty in J_{ℓ} and u (0, T_{ℓ}) = 0 & if ξ_{i_{ℓ}} = 2 . \end{matrix} \end{array}$ In particular, $T_{1}, \dots, T_{q} \in T$ . Moreover, in view of (9.41)–(9.42), we actually have $\begin{array}{l} T_{ℓ} \in int (J_{ℓ}), ℓ \in {1, \dots, q} . \end{array}$ Since $T$ is finite by Theorem 1.3(i), each $T_{ℓ}$ has a neighborhood containing no other point of $T$ and it follows from Theorem 1.3(ii) that, for some small $ε > 0$ , $\begin{array}{l} (9.74) & \{\begin{matrix} u is classical on [T_{ℓ} - ε, T_{ℓ} + ε] ∖ {T_{ℓ}} & if ξ_{i_{ℓ}} = 0 \\ u (0, \cdot) > 0 in [T_{ℓ} - ε, T_{ℓ} + ε] ∖ {T_{ℓ}} & if ξ_{i_{ℓ}} = 2, \end{matrix} \end{array}$ i.e. $T_{ℓ}$ is a GBU time without LBC if $ξ_{i_{ℓ}} = 0$ and a bouncing time if $ξ_{i_{ℓ}} = 2$ . In what follows we denote $\begin{array}{l} Θ = {T_{1}, \dots, T_{q}} = Θ_{0} \cup Θ_{2}, \end{array}$ where $\begin{array}{l} (9.75) & Θ_{0} = {t \in Θ; u is classical on [t - ε, t + ε] ∖ {t}} \end{array}$ and $\begin{array}{l} (9.76) & Θ_{2} = {t \in Θ; u (0, \cdot) > 0 in [t - ε, t + ε] ∖ {t}} . \end{array}$

In order to conclude, it remains to enumerate the other elements of $T$ , by counting the corresponding drops of the function $N (t)$ . Recall that the indices $κ_{1}, \dots, κ_{d + 1}$ are defined in (9.25). We claim that for each $j \in {1, \dots, k}$ , the interval $({\hat{s}}_{κ_{j}}, {\hat{s}}_{κ_{j + 1}})$ contains at least $\begin{array}{l} (9.77) & \{\begin{matrix} 1 element of Θ_{0} & if ξ_{κ_{j}} = 0 \\ 2 elements of T ∖ Θ & if ξ_{κ_{j}} = 1 \\ (κ_{j + 1} - κ_{j} - 1) elements of Θ_{2} and 2 elements of T ∖ Θ & if ξ_{κ_{j}} = 2 \end{matrix} \end{array}$ (for the example from Fig. 8, this is illustrated in Fig. 11).

Fig. 11.

Illustration of property (9.77) for the example in Fig. 8 (here $κ_{1} = 1, κ_{2} = 2, κ_{3} = 5, κ_{4} = 6$ ). The bullets ∙ mark additional times in $T ∖ Θ$ .

Indeed, the first case in (9.77) is guaranteed by (9.73)–(9.74) and the second case by (9.41)–(9.42) and Theorem 1.3(ii). To check the last case, by (9.26), we see that for each $κ \in {κ_{j}, \dots, κ_{j + 1} - 2}$ the interval $(s_{κ}, s_{κ + 1})$ contains at least one element of $Θ_{2}$ , by (9.73)–(9.74), whereas each of the intervals $({\hat{s}}_{κ_{j}}, s_{κ_{j}})$ and $(s_{κ_{j + 1} - 1}, {\hat{s}}_{κ_{j + 1}})$ contains at least one element of $T ∖ Θ$ by (9.41)–(9.42) and Theorem 1.3(ii).

Now, by Proposition 7.2, (9.30) and (9.77), for all $j \in {1, \dots, d}$ , we have $\begin{array}{l} (9.78) & N ({\hat{s}}_{κ_{j}}) - N ({\hat{s}}_{κ_{j + 1}}) ⩾ \{\begin{matrix} 2 & if ξ_{κ_{j}} = 0 or 1 \\ 2 (κ_{j + 1} - κ_{j} - 1) + 2 & if ξ_{κ_{j}} = 2 \end{matrix} \end{array}$ hence $\begin{array}{l} N ({\hat{s}}_{κ_{j}}) - N ({\hat{s}}_{κ_{j + 1}}) ⩾ 2 (κ_{j + 1} - κ_{j}) . \end{array}$ Therefore, $\begin{array}{l} N ({\hat{s}}_{1}) - N ({\hat{s}}_{m + 1}) = N ({\hat{s}}_{κ_{1}}) - N ({\hat{s}}_{κ_{d + 1}}) ⩾ 2 (κ_{d + 1} - κ_{1}) = 2 m . \end{array}$ But $N (0) ⩽ 2 m$ , due to (9.39), and $N (t) \in N$ is nonincreasing by Proposition 6.1(ii) and (9.30). Consequently, the inequality in (9.78) is actually an equality. For all $j \in {1, \dots, d}$ , setting $I_{j} : = ({\hat{s}}_{κ_{j}}, {\hat{s}}_{κ_{j + 1}})$ , it then follows from (9.77) and Proposition 7.2 that $T \subset ({\hat{s}}_{m}, {\hat{s}}_{0})$ and that $\begin{array}{l} I_{j} \cap T consists of exactly \{\begin{matrix} 1 element of Θ_{0}, & if ξ_{κ_{j}} = 0 \\ 2 elements of T ∖ Θ, & if ξ_{κ_{j}} = 1 \\ (κ_{j + 1} - κ_{j} - 1) elements of Θ_{2} \\ and 2 elements of T ∖ Θ, & if ξ_{κ_{j}} = 2 . \end{matrix} \end{array}$ On the other hand, by (9.26), we have $ξ_{κ_{j}} = {\bar{σ}}_{j}$ if ${\bar{σ}}_{j} ⩽ 1$ , whereas $ξ_{κ_{j}} = 2$ and $κ_{j + 1} - κ_{j} = {\bar{σ}}_{j}$ otherwise. Applying Theorem 1.3(ii), recalling (9.41), (9.75), (9.76), it follows that $\begin{array}{l} \{\begin{matrix} u is classical in I_{j} except for a single time in Θ_{0}, & if {\bar{σ}}_{j} = 0 \\ {t \in I_{j}, u (0, t) > 0} is a nonempty open subinterval of I_{j}, & if {\bar{σ}}_{j} = 1 \\ {t \in I_{j}, u (0, t) > 0} is a nonempty open subinterval of I_{j} \\ minus {\bar{σ}}_{j} - 1 elements of Θ_{2}, & if {\bar{σ}}_{j} ⩾ 2 . \end{matrix} \end{array}$ Consequently, the assertion in Theorem 1.4’ is satisfied with ${\bar{t}}_{j} = {\hat{s}}_{κ_{j}}$ for $j = 1, \dots$ , $d + 1$ . As for property (9.1), it follows from (9.33), (9.34). The proof is complete. □

Footnotes

Appendix

We here prove Proposition 9.2. The proof of assertion (ii) will use the following simple lemma.

Acknowledgements

NM is supported by the JSPS Grant-in-Aid for Scientific Research (B) (No.20H01814). PhS is partially supported by the Labex MME-DII (ANR11-LBX-0023-01).

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Singularity formation and regularization at multiple times in the viscous Hamilton–Jacobi equation

Abstract

Keywords

1. Introduction and main results

1.1. Problem

1.2. Applications to stochastic control problems

1.3. Results in general domains

Remark 1.2 (Radial case in higher dimensions).

Remark 1.3 (Further development).

1.5. Differences from Fujita equation

1.6. Ideas of proofs

1 Although the above description is more convenient for heuristic purposes, the actual construction is done in the converse direction, first constructing the bump which is farthest from the boundary.

Footnotes

Appendix

Acknowledgements

References

¹
Although the above description is more convenient for heuristic purposes, the actual construction is done in the converse direction, first constructing the bump which is farthest from the boundary.