Exponential grow-up rates in a quasilinear Keller

Abstract

The chemotaxis system $\begin{matrix} (⋆) & \{\begin{matrix} u_{t} = \nabla \cdot (D (u) \nabla u) - \nabla \cdot (u S (u) \nabla v), \\ 0 = Δ v - μ + u, μ = \frac{1}{| Ω |} \int_{Ω} u, \end{matrix} \end{matrix}$ is considered in a ball $Ω = B_{R} (0) \subset R^{n}$ .

It is shown that if $S \in C^{2} ([0, \infty))$ suitably generalizes the prototype given by $\begin{array}{rcl} S (ξ) = \frac{χ}{ξ + 1}, ξ ⩾ 0, \end{array}$ with some $χ > 0$ , and if diffusion is suitably weak in the sense that $0 < D \in C^{2} ((0, \infty))$ is such that there exist $K_{D} > 0$ and $m \in (- \infty, 1 - \frac{2}{n})$ fulfilling $\begin{array}{rcl} D (ξ) ⩽ K_{D} ξ^{m - 1} for all ξ > 0, \end{array}$ then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution $(u, v)$ which blows up in infinite time and satisfies $\begin{array}{rcl} \frac{1}{C} e^{χ t} ⩽ {‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} ⩽ C e^{χ t} for all t > 0 . \end{array}$ A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).

Keywords

Chemotaxis singularity formation grow-up rate

1. Introduction

Within the class of Keller–Segel type problems ([22]) given by $\begin{matrix} (1.1) & \{\begin{matrix} u_{t} = \nabla \cdot (D (u) \nabla u) - \nabla \cdot (u S (u) \nabla v), & x \in Ω, t > 0, \\ 0 = Δ v - μ + u, μ = \frac{1}{| Ω |} \int_{Ω} u, \int_{Ω} v = 0, & x \in Ω, t > 0, \\ (D (u) \nabla u - u S (u) \nabla v) \cdot ν = 0, \nabla v \cdot ν = 0, & x \in \partial Ω, t > 0, \\ u (x, 0) = u_{0} (x), & x \in Ω, \end{matrix} \end{matrix}$ systems involving asymptotically constant chemotactic sensitivity functions $0 ⩽ ξ \mapsto ξ S (ξ)$ , such as that quantified by the choice $\begin{matrix} (1.2) & S (ξ) = \frac{χ}{ξ + 1}, ξ ⩾ 0, \end{matrix}$ play a prominent role in two respects. On the one hand, such selections constitute the apparently most straightforward option to account for saturation effects in chemotactic responses which depend on the population density u, as underlying diverse approaches in the literature concerned with modeling of attractive taxis in various concrete application frameworks (see [24,31] and [27] for some examples, and also the discussions in [35] and [16]).

On the other hand, the choice in (1.2) corresponds to a borderline situation from a mathematical perspective: In the context of general power-type asymptotic behavior of its nonlinear ingredients, as prototypically represented by the selections $\begin{matrix} (1.3) & D (ξ) = K_{D} {(ξ + ι)}^{m - 1} and S (ξ) = χ {(ξ + 1)}^{- ϑ - 1}, ξ > 0, \end{matrix}$ namely, in the case when ϑ is positive, the problem (1.1), for simplicity here assumed to be posed in a ball $Ω \subset R^{n}$ with $n ⩾ 2$ , can be seen to admit global classical solutions for arbitrary $m \in R$ , $ι ⩾ 0$ , $χ > 0$ , $μ > 0$ and any initial data which are such that $\begin{matrix} (1.4) & u_{0} \in W^{1, \infty} (Ω) is positive in \overline{Ω} and radially symmetric with \frac{1}{| Ω |} \int_{Ω} u_{0} d x = μ \end{matrix}$ ([39, Proposition 1.2]); if $ϑ < 0$ , however, then a conclusion of this flavor has been drawn only when additionally the considered diffusion mechanism is strong enough in the sense that $m + ϑ > 1 - \frac{2}{n}$ , whereas in the case when $m + ϑ < 1 - \frac{2}{n}$ , for each $μ > 0$ one can find initial data fulfilling (1.4) such that (1.1) possesses a solution blowing up in finite time ([5,6]; cf. also [2] for a blow-up result for $m + λ = 1 - \frac{2}{n}$ in the particular case when $ϑ = - 1$ ). Beyond these findings concerned with the comparatively simple model (1.1), the literature of the past few years has moreover collected some indications for a similar borderline role of (1.2) also in the broader context of more complex Keller–Segel systems, such as a fully parabolic variant of (1.1) ([3,4,18,23,34,37]).

Main results: Exponential grow-up in the presence of borderline sensitivities. The present manuscript now aims at describing possible facets of solution behavior in (1.1) in situations suitably generalizing the borderline-type case determined by (1.2). To substantiate our goal in this regard, we note that in the context of (1.1)-(1.3), irrespective of the sign of ϑ the condition $m + ϑ > 1 - \frac{2}{n}$ ensures that the mentioned global classical solutions in fact remain bounded throughout evolution ([6,18,34]), whereas if $m + ϑ < 1 - \frac{2}{n}$ , then some unbounded solutions always exist both when $ϑ > 0$ and when $ϑ < 0$ ([6,39]). In conjunction with the above, this suggests to expect a highlighted role of the value $ϑ = 0$ with respect to unboundedness phenomena only in the part of the $(m, ϑ)$ plane where $m + ϑ < 1 - \frac{2}{n}$ .

Thus focusing on this region henceforth, we furthermore recall that for positive ϑ, despite the mentioned result on unconditional global solvability, it is known that some of these solutions $(u, v)$ blow up in infinite time, at a rate determined by the two-sided inequality $\begin{matrix} (1.5) & \frac{1}{C} {(t + 1)}^{\frac{1}{ϑ}} ⩽ {‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} ⩽ C {(t + 1)}^{\frac{1}{ϑ}}, \end{matrix}$ valid for all $t > 0$ with some $C = C (u, v) > 0$ ; in particular, grow-up at arbitrarily fast rates can be achieved on choosing $ϑ > 0$ suitably small.

The intention of the present study now consists in examining in a quantitative manner how the transition from such algebraic infinite-time blow-up for $ϑ > 0$ on the one hand, and finite-time explosions for $ϑ < 0$ on the other, is realized in the separating point $ϑ = 0$ . Our main results in this direction detect exponentially fast grow-up in this borderline case, at rates which can quite precisely be specified in dependence on the asymptotics of S, and which in the particular situation from (1.2) are hence exclusively determined by the parameter χ therein.

The first and major part of this outcome, actually allowing for choices of D and S somewhat more general than those in (1.3) and (1.2), asserts the occurrence of infinite-time blow-up and provides an estimate from below for the corresponding rate.

Theorem 1.1.
Let $n ⩾ 2$ , $R > 0$ and $Ω = B_{R} (0) \subset R^{n}$ , and suppose that $\begin{matrix} (1.6) & D \in C^{2} ((0, \infty)) is positive, \end{matrix}$ that $\begin{matrix} (1.7) & S \in C^{2} ([0, \infty)) is such that \underset{ξ \to \infty}{lim sup} {ξ S (ξ)} < \infty, \end{matrix}$ and that moreover there exist $ξ_{0} > 0$ , $K_{D} > 0$ , $K_{S} > 0$ , $χ > 0$ , $λ \in (1, 2]$ and $\begin{matrix} (1.8) & m \in (- \infty, 1 - \frac{2}{n}) \end{matrix}$ such that $\begin{matrix} (1.9) & D (ξ) ⩽ K_{D} \cdot ξ^{m - 1} for all ξ ⩾ ξ_{0} \end{matrix}$ and $\begin{matrix} (1.10) & S (ξ) ⩾ \frac{χ}{ξ} - \frac{K_{S}}{ξ^{λ}} for all ξ ⩾ ξ_{0} . \end{matrix}$ Then there exists $μ_{⋆} > 0$ such that for any choice of $μ > μ_{⋆}$ , one can find radially symmetric initial data $u_{0}$ such that ( 1.4 ) holds, and such that ( 1.1 ) admits a unique global classical solution $(u, v)$ , with $\begin{matrix} (1.11) & \{\begin{matrix} u \in C^{0} (\overline{Ω} \times [0, \infty)) \cap C^{2, 1} (\overline{Ω} \times (0, \infty)) and \\ v \in C^{2, 0} (\overline{Ω} \times (0, \infty)), \end{matrix} \end{matrix}$ for which we have $u > 0$ in $\overline{Ω} \times [0, \infty)$ , and for which we can find $C > 0$ in such a way that $\begin{matrix} (1.12) & {‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} ⩾ C e^{χ t} for all t ⩾ 0 . \end{matrix}$
Remark.
Our argument will actually show that for each $μ > μ_{⋆}$ one can find a nondecreasing function $M = M^{(μ)} \in C^{0} ([0, R])$ which is such that ${sup}_{r \in (0, R)} \frac{M (r)}{r^{n}} < \infty$ and $M (R) = μ | Ω |$ , and that the above conclusion in fact holds for any $u_{0}$ fulfilling (1.4) as well as $\begin{array}{rcl} \int_{B_{r} (0)} u_{0} d x ⩾ M (r) for all r \in (0, R) . \end{array}$ As this can readily be seen to be satisfied, for instance, by all radially nonincreasing $u_{0} \in C^{1} (\overline{Ω})$ which beyond (1.4) satisfy $supp u_{0} \subset B_{r_{⋆}} (0)$ with some suitably small $r_{⋆} = r_{⋆} (M) \in (0, R)$ , the behavior described by Theorem 1.1 is in fact evinced by a considerably large group of solutions to (1.1).

Optimality of (1.12) is now confirmed by a second statement which, in fact, can be verified without any restrictions on the spatial dimension nor any requirements on the diffusion rate D which go beyond (1.6).
Proposition 1.2.
Let $Ω = B_{R} (0) \subset R^{n}$ with some $n ⩾ 1$ and $R > 0$ , and suppose that besides ( 1.6 ) and ( 1.7 ), we have $\begin{matrix} (1.13) & S (ξ) ⩽ \frac{χ}{ξ} + \frac{K_{S}}{ξ^{λ}} for all ξ ⩾ ξ_{0} \end{matrix}$ with some $ξ_{0} > 0$ , $χ > 0$ , $K_{S} > 0$ and $λ > 1$ . Then for any $μ > 0$ and each $u_{0}$ fulfilling ( 1.4 ), the problem ( 1.1 ) possesses a unique global classical solution $(u, v)$ satisfying ( 1.11 ) and $u > 0$ in $\overline{Ω} \times [0, \infty)$ , and this solution satisfies $\begin{matrix} (1.14) & {‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} ⩽ C e^{χ t} for all t ⩾ 0 \end{matrix}$ with some $C > 0$ .

Upon the simple observation that $\frac{1}{ξ} ⩾ \frac{1}{ξ + 1} = \frac{1}{ξ} - \frac{1}{ξ (ξ + 1)} ⩾ \frac{1}{ξ} - \frac{1}{ξ^{2}}$ for all $ξ > 0$ , we thus particularly infer that if $n ⩾ 2$ and $R > 0$ , and if with some $m \in (- \infty, 1 - \frac{2}{n})$ , $ι ⩾ 0$ and $χ > 0$ we choose D as in (1.3) and S as in (1.2), then the accordingly obtained version of (1.1) indeed admits some global unbounded solution satisfying both (1.12) and (1.14).

It may be remarked that comparably far-reaching descriptions of blow-up mechanisms in systems of type (1.1), but also in related Keller–Segel systems, have been achieved only in quite a limited number of situations so far. Rates of infinite-time blow-up have been precisely determined for some precisely mass-critical solutions to a Neumann-Dirichlet type variant of (1.1) with $D \equiv S \equiv 1$ ([21]), while further grow-up phenomena in certain relatives of (1.1) have so far been described mainly qualitatively, with information on respective rates at most available as one-sided upper estimates ([1,3,10,36]). Likewise, quantitative knowledge on finite-time singularity formation has been achieved only in a comparatively small selection of cases, both with regard to corresponding rates ([11–15,19,26,28,29]), and with respect to spatial behavior near blow-up ([7–9,17,25,30,32,33,38]).

Plan of the paper. After a collation of some known preliminaries in Section 2, Section 3 will be devoted to the construction of candidates for subsolutions $\underline{w}$ to be used in the course of a comparison argument for a parabolic problem satisfied by the mass accumulation function which, given a global radial solution $(u, v)$ of (1.1), is defined in reminiscence of the classical precedent [20] by letting $w (s, t) : = \frac{1}{n | B_{1} (0) |} \int_{B_{s^{1 / n}} (0)} u (x, t) d x$ for $s \in [0, R^{n}]$ and $t ⩾ 0$ (Lemma (2.2)). Here a major challenge will consist in the identification of a subregion of $[0, R^{n}] \times [0, \infty)$ within which an alteration between expectedly steep gradients of these functions $\underline{w}$ near the origin and essentially flat structure near the spatial boundary can be described through a functional form which appropriately captures the anticipated behavior of the unknown solution. As it will turn out in Lemma 3.2 and in the core of our analysis contained in Section 4, by suitably designed subsolutions to said problem such a transition can be achieved in annular regions of the form $B_{r_{1} (t)} (0) ∖ B_{r_{0} (t)} (0)$ with radii exhibiting exponential decay in the sense that $\begin{array}{rcl} r_{0} (t) \approx c_{1} e^{- \frac{χ}{n} \cdot t} and r_{1} (t) \approx c_{2} e^{- η t}, t ≃ \infty, \end{array}$ with some $c_{1} > 0$ , $c_{2} > 0$ and $η > 0$ (cf. (3.9) and (3.4)). Mainly on the basis of this construction, in Section 5 the statements from Theorem 1.1 and Proposition 1.2 will be accomplished by means of two comparison arguments.
2. Global existence and transformation to a scalar problem

In this preliminary section we formulate three basic statements concerned with global solvability, a transformation to a scalar parabolic problem, and a comparison principle for the latter. As corresponding verifications can be achieved by verbatim copying corresponding arguments from [39, Section 2], we may refrain from repeating proofs here.

Firstly, thanks to the requirement on decay of S contained therein the assumptions in (1.6) and (1.7) ensure unique global solvability of (1.1) in the following sense ([39, Lemma 2.1]).

Lemma 2.1.
Let $n ⩾ 1$ , $R > 0$ and $Ω = B_{R} (0) \subset R^{n}$ , let $μ > 0$ , and assume that D, S and $u_{0}$ satisfy ( 1.6 ), ( 1.7 ) and ( 1.4 ) with $\frac{1}{| Ω |} \int_{Ω} u_{0} d x = μ$ . Then there exist uniquely determined functions u and v fulfilling ( 1.11 ) and such that $u > 0$ in $\overline{Ω} \times [0, \infty)$ , that $(u (\cdot, t), v (\cdot, t))$ is radially symmetric with respect to $x = 0$ for all $t > 0$ , and that $(u, v)$ solves ( 1.1 ) in the classical sense. Moreover, $\begin{matrix} (2.1) & \int_{Ω} u (x, t) d x = \int_{Ω} u_{0} d x for all t > 0 . \end{matrix}$

We next summarize some basic features associated with a variable transformation which actually dates back at least to [20] (cf. [39, Lemma 2.2]).
Lemma 2.2.
Let $n ⩾ 1$ , $R > 0$ and $Ω = B_{R} (0) \subset R^{n}$ , let $μ > 0$ , and let ( 1.6 ), ( 1.7 ) and ( 1.4 ) be fulfilled with $\frac{1}{| Ω |} \int_{Ω} u_{0} d x = μ$ . Then letting $\begin{matrix} (2.2) & w (s, t) : = \int_{0}^{s^{\frac{1}{n}}} r^{n - 1} u (r, t) d r, s \in [0, R^{n}], t ⩾ 0, \end{matrix}$ we obtain a nonnegative function $w \in C^{0} ([0, \infty); C^{1} ([0, R^{n}])) \cap C^{2, 1} ([0, R^{n}] \times (0, \infty))$ which satisfies $\begin{matrix} (2.3) & w_{s} (s, t) = \frac{1}{n} u (s^{\frac{1}{n}}, t) for all s \in [0, R^{n}] and t > 0 \end{matrix}$ as well as $\begin{matrix} (2.4) & w (0, t) = 0 and w (R^{n}, t) = \frac{μ R^{n}}{n} for all t > 0, \end{matrix}$ and moreover $\begin{matrix} (2.5) & (P w) (s, t) = 0 for all s \in (0, R^{n}) and t > 0 . \end{matrix}$ Here, for $T > 0$ , $s_{1} ⩾ 0$ , $s_{2} > s_{1}$ and positive functions $φ \in C^{1} ((s_{1}, s_{2}) \times (0, T))$ which for all $t \in (0, T)$ satisfy $φ_{s} (\cdot, t) > 0$ in $(s_{1}, s_{2})$ and $φ (\cdot, t) \in C^{2} ((s_{1}, s_{2}) ∖ N (t))$ with some finite set $N (t) \subset R$ , we have set $\begin{array}{rcl} (P φ) (s, t) & : = & φ_{t} (s, t) - n^{2} s^{2 - \frac{2}{n}} D (n φ_{s} (s, t)) φ_{s s} (s, t) \\ (2.6) & - {φ (s, t) - \frac{μ}{n} s} \cdot S (n φ_{s} (s, t)) \cdot n φ_{s} (s, t) \end{array}$ for $t \in (0, T)$ and $s \in (s_{1}, s_{2}) ∖ N (t)$ .

A final preparation documents a comparison principle suitably arranged in such a way that the diffusion degeneracy in (2.5) at $s = 0$ can adequately be coped with, as well as jump-type discontinuities in second order spatial derivatives which will arise in our construction of subsolutions during the next section. A proof can be found detailed in ([39, Lemma 2.3]). Lemma 2.3.
Let $n ⩾ 1$ , $R > 0$ and $Ω = B_{R} (0) \subset R^{n}$ , and suppose that $D \in C^{0} ((0, \infty))$ and $S \in C^{0} ([0, \infty))$ , that $μ > 0$ and $T \in (0, \infty]$ , and that $\underline{w}$ and $\overline{w}$ belong to $C^{0} ([0, T); C^{1} ([0, R^{n}]) \cap C^{1} ((0, R^{n}) \times (0, T))$ and are such that $\begin{array}{rcl} {\underline{w}}_{s} (s, t) > 0 and \overline{w} (s, t) > 0 for all s \in [0, R^{n}] and t \in [0, T) \end{array}$ as well as $\begin{array}{rcl} \underline{w} (\cdot, t) \in W_{loc}^{2, \infty} ((0, R^{n})) and \overline{w} (\cdot, t) \in W_{loc}^{2, \infty} ((0, R^{n})) for all t \in (0, T) . \end{array}$ Then under the assumption that for all $t \in (0, T)$ and each common differentiability point $s \in (0, R^{n})$ of ${\underline{w}}_{s} (\cdot, t)$ and ${\overline{w}}_{s} (\cdot, t)$ we have $\begin{array}{rcl} {\underline{w}}_{t} ⩽ n^{2} s^{2 - \frac{2}{n}} D (n {\underline{w}}_{s}) {\underline{w}}_{s s} + (\underline{w} - \frac{μ}{n} s) \cdot S (n {\underline{w}}_{s}) \cdot n {\underline{w}}_{s} \end{array}$ and $\begin{array}{rcl} {\overline{w}}_{t} ⩾ n^{2} s^{2 - \frac{2}{n}} D (n {\overline{w}}_{s}) {\overline{w}}_{s s} + (\overline{w} - \frac{μ}{n} s) \cdot S (n {\overline{w}}_{s}) \cdot n {\overline{w}}_{s}, \end{array}$ and that furthermore $\begin{array}{rcl} \underline{w} (0, t) ⩽ \overline{w} (0, t) and \underline{w} (R^{n}, t) ⩽ \overline{w} (R^{n}, t) for all t \in (0, T) \end{array}$ as well as $\begin{array}{rcl} \underline{w} (s, 0) ⩽ \overline{w} (s, 0) for all s \in (0, R^{n}), \end{array}$ it follows that $\begin{array}{rcl} \underline{w} (s, t) ⩽ \overline{w} (s, t) for all s \in [0, R^{n}] and t \in [0, T) . \end{array}$

3. A three-parameter family of comparison functions

To launch our construction of candidates for subsolutions to the parabolic problem associated with (2.4)-(2.6), we first rely on our overall assumptions on the system constituents made in Theorem 1.1 in choosing some parameters that will determine essential parts of our approach.

Lemma 3.1.
Let $n ⩾ 2$ , $m \in (- \infty, 1 - \frac{2}{n})$ , $λ \in (1, 2]$ and $χ > 0$ . Then there exist $γ > 1$ , $κ \in (\frac{γ - 1}{γ}, 1)$ , $δ \in (0, 1]$ and $ρ \in (0, χ)$ such that $\begin{matrix} (3.1) & - m γ + 1 - \frac{2}{n} > 0 \end{matrix}$ and $\begin{matrix} (3.2) & δ ⩽ m (γ - 1) + κ \cdot (- m γ + 1 - \frac{2}{n}) \end{matrix}$ as well as $\begin{matrix} (3.3) & ρ ⩾ max {\frac{χ}{γ}, {1 - (1 - γ + κ γ) (λ - 1)} \cdot χ, (1 - δ) χ} . \end{matrix}$
Proof.
While, regardless of the sign of m, the existence of γ fulfilling (3.1) is obvious, using this inequality we may thereafter fix actually an arbitrary $κ \in (\frac{γ - 1}{γ}, 1)$ to infer that then, thanks to our hypothesis that $n ⩾ 2$ , $\begin{array}{rcl} m (γ - 1) + κ \cdot (- m γ + 1 - \frac{2}{n}) > m (γ - 1) + \frac{γ - 1}{γ} \cdot (- m γ + 1 - \frac{2}{n}) = \frac{γ - 1}{γ} \cdot (1 - \frac{2}{n}), \end{array}$ so that $\begin{array}{rcl} δ : = min {1, m (γ - 1) + κ \cdot (- m γ + 1 - \frac{2}{n})} \end{array}$ indeed belongs to $(0, 1]$ . The proof can be completed by observing that thus $\frac{1}{γ} < 1$ and $2 - λ < 1$ as well as $1 - δ < 1$ , whence simply setting $ρ : = max {\frac{χ}{γ}, {1 - (1 - γ + κ γ) (λ - 1)} \cdot χ, (1 - δ) χ}$ we obtain $ρ \in (0, χ)$ fulfilling (3.3). □

Besides the numbers γ, κ, δ and ρ that have been introduced above and that will be kept fixed throughout the sequel, our construction will involve two further parameters a and b which, along with the mass level determined by quantity μ, will remain at our free disposal up to the final step achieved in the proof of Theorem 1.1. Specifically, the functions to be identified as subsolutions for the parabolic operator in (2.6) will be of the structure described as follows.
Lemma 3.2.
Suppose that $n ⩾ 2$ , $R > 0$ , $m \in (- \infty, 1 - \frac{2}{n})$ , $χ > 0$ and $λ \in (1, 2]$ , and let γ, δ and ρ be as introduced in Lemma 3.1 . Then one can find $a_{0} ⩾ 1$ with the property that whenever $μ > 0$ , $a > a_{0}$ and $b > 0$ , writing $\begin{matrix} (3.4) & y (t) \equiv y^{(a, b)} (t) : = a e^{χ t} + b e^{ρ t}, t ⩾ 0, \end{matrix}$ and $\begin{matrix} (3.5) & θ (t) \equiv θ^{(μ, a, b)} (t) : = \frac{\frac{μ R^{n}}{n}}{\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ - 1}{γ}} (t)}, t ⩾ 0, \end{matrix}$ and letting $\begin{matrix} (3.6) & {\underline{w}}_{in} (s, t) \equiv {\underline{w}}_{in}^{(μ, a, b)} (s, t) : = θ (t) y (t) s, s ⩾ 0, t ⩾ 0, \end{matrix}$ and $\begin{matrix} (3.7) & {\underline{w}}_{mid} (s, t) \equiv {\underline{w}}_{mid}^{(μ, a, b)} (s, t) : = \frac{γ}{γ - 1} θ (t) - \frac{1}{γ - 1} θ (t) y^{1 - γ} (t) s^{1 - γ}, s > 0, t ⩾ 0, \end{matrix}$ and $\begin{matrix} (3.8) & {\underline{w}}_{out} (s, t) \equiv {\underline{w}}_{out}^{(μ, a, b)} (s, t) : = \frac{μ R^{n}}{n} - θ (t) \cdot (R^{n} - s), s ⩾ 0, t ⩾ 0, \end{matrix}$ as well as $\begin{matrix} (3.9) & \underline{w} (s, t) \equiv {\underline{w}}^{(μ, a, b)} (s, t) : = \{\begin{matrix} {\underline{w}}_{in} (s, t), & t ⩾ 0, s \in [0, \frac{1}{y (t)}), \\ {\underline{w}}_{mid} (s, t), & t ⩾ 0, s \in [\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}), \\ {\underline{w}}_{out} (s, t), & t ⩾ 0, s \in [\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n}], \end{matrix} \end{matrix}$ we obtain nonnegative functions $y \in C^{1} ([0, \infty))$ , $θ \in C^{1} ([0, \infty))$ and $\underline{w} \in C^{1} ([0, R^{n}] \times [0, \infty))$ such that $\underline{w} (\cdot, t) \in C^{2} ([0, R^{n}] ∖ N (t))$ with $N (t) : = {\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}}$ for all $t > 0$ , that $\underline{w} (0, t) = 0$ and $\underline{w} (R^{n}, t) = \frac{μ R^{n}}{n}$ for all $t ⩾ 0$ , and that $\begin{matrix} (3.10) & θ (t) ⩾ θ_{\infty} for all t > 0 \end{matrix}$ and $\begin{matrix} (3.11) & {\underline{w}}_{s} (s, t) ⩾ θ_{\infty} for all s ⩾ 0 and t ⩾ 0, \end{matrix}$ where $\begin{matrix} (3.12) & θ_{\infty} \equiv θ_{\infty}^{(μ)} : = \frac{\frac{μ R^{n}}{n}}{\frac{γ}{γ - 1} + R^{n}} . \end{matrix}$ Moreover, $\begin{matrix} (3.13) & θ^{'} (t) = - \frac{\frac{μ R^{n}}{n}}{{[\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ - 1}{γ}} (t)]}^{2}} \cdot y^{\frac{1}{γ} - 2} (t) y^{'} (t) for all t > 0, \end{matrix}$ and besides $\begin{matrix} (3.14) & \{\begin{matrix} \partial_{t} {\underline{w}}_{i n} (s, t) = θ^{'} (t) y (t) s + θ (t) y^{'} (t) s, \\ \partial_{s} {\underline{w}}_{i n} (s, t) = θ (t) y (t) and \\ \partial_{s}^{2} {\underline{w}}_{i n} (s, t) = 0, \end{matrix} \end{matrix}$ we have $\begin{matrix} (3.15) & \{\begin{matrix} \partial_{t} {\underline{w}}_{m i d} (s, t) = \frac{γ}{γ - 1} θ^{'} (t) - \frac{1}{γ - 1} θ^{'} (t) y^{1 - γ} (t) s^{1 - γ} + θ (t) y^{- γ} (t) y^{'} (t) s^{1 - γ}, \\ \partial_{s} {\underline{w}}_{m i d} (s, t) = θ (t) y^{1 - γ} (t) s^{- γ} and \\ \partial_{s}^{2} {\underline{w}}_{m i d} (s, t) = - γ θ (t) y^{1 - γ} (t) s^{- γ - 1} \end{matrix} \end{matrix}$ as well as $\begin{matrix} (3.16) & \{\begin{matrix} \partial_{t} {\underline{w}}_{o u t} (s, t) = - θ^{'} (t) \cdot (R^{n} - s), \\ \partial_{s} {\underline{w}}_{o u t} (s, t) = θ (t) and \\ \partial_{s}^{2} {\underline{w}}_{o u t} (s, t) = 0 \end{matrix} \end{matrix}$ for all $s > 0$ and $t > 0$ .
Proof.
We let $\begin{array}{rcl} a_{0} : = max {1, R^{- \frac{n γ}{γ - 1}}}, \end{array}$ and for fixed $a > a_{0}$ and $b ⩾ 0$ we take y as accordingly introduced in (3.4). Then $y (t) ⩾ a ⩾ 1$ and hence $\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ}{γ - 1}} (t) ⩾ R^{n}$ for all $t \in [0, \infty)$ , so that (3.5) indeed determines a nonnegative function $θ \in C^{1} ([0, \infty))$ fulfilling (3.13). For ${\underline{w}}_{i n}$ , ${\underline{w}}_{m i d}$ and ${\underline{w}}_{out}$ as correspondingly defined through (3.6), (3.7) and (3.8), the verification of (3.14), (3.15) and (3.16) can thereupon be achieved by straightforward computation.

Noting that the restriction $a > a_{0} ⩾ R^{- \frac{n γ}{γ - 1}}$ together with the inequality $y ⩾ a > 1$ guarantees that $\frac{1}{y (t)} < \frac{1}{y^{(γ - 1) / γ} (t)} < R^{n}$ for all $t > 0$ , we thus infer that (2.2) indeed provides a function $\underline{w}$ on $[0, R^{n}] \times [0, \infty)$ in a well-defined manner, and that $\underline{w} (0, t) = {\underline{w}}_{in} (0, t) = 0$ and $\underline{w} (R^{n}, t) = {\underline{w}}_{out} (R^{n}, t) = \frac{μ R^{n}}{n}$ for all $t ⩾ 0$ . Since clearly $\begin{array}{rcl} {\underline{w}}_{mid} (\frac{1}{y (t)}, t) = \frac{γ}{γ - 1} θ (t) - \frac{1}{γ - 1} θ (t) = θ (t) = {\underline{w}}_{in} (\frac{1}{y (t)}, t) for all t ⩾ 0 \end{array}$ and $\begin{array}{rcl} \partial_{s} {\underline{w}}_{mid} (\frac{1}{y (t)}, t) = θ (t) y^{1 - γ} (t) y^{γ} (t) = θ (t) y (t) = \partial_{s} {\underline{w}}_{in} (\frac{1}{y (t)}, t) for all t ⩾ 0 \end{array}$ as well as $\begin{array}{rcl} \partial_{s} {\underline{w}}_{mid} (\frac{1}{y^{(γ - 1) / γ} (t)}, t) & = & θ (t) y^{1 - γ} (t) \cdot {(y^{- \frac{γ - 1}{γ}} (t))}^{- γ} \\ = & θ (t) = \partial_{s} {\underline{w}}_{out} (\frac{1}{y^{(γ - 1) / γ} (t)}, t) for all t ⩾ 0, \end{array}$ and since our selection of θ ensures that $\begin{array}{l} {\underline{w}}_{mid} (\frac{1}{y^{\frac{γ - 1}{γ}} (t)}, t) - {\underline{w}}_{out} (\frac{1}{y^{\frac{γ - 1}{γ}} (t)}, t) \\ = \frac{γ}{γ - 1} θ (t) - \frac{1}{γ - 1} θ (t) y^{1 - γ} (t) \cdot {(y^{- \frac{γ - 1}{γ}} (t))}^{1 - γ} \\ - {\frac{μ R^{n}}{n} - θ (t) \cdot (R^{n} - y^{- \frac{γ - 1}{γ}} (t))} \\ = \frac{γ}{γ - 1} θ (t) - \frac{1}{γ - 1} θ (t) y^{- \frac{γ - 1}{γ}} (t) \\ - \frac{μ R^{n}}{n} + R^{n} θ (t) - θ (t) y^{- \frac{γ - 1}{γ}} (t) \\ = {\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ - 1}{γ}} (t)} \cdot θ (t) - \frac{μ R^{n}}{n} \\ = 0 for all t ⩾ 0, \end{array}$ it moreover follows that $\underline{w}$ and ${\underline{w}}_{s}$ are continuous on $[0, R^{n}] \times [0, \infty)$ , whereas continuity also of ${\underline{w}}_{t}$ can similarly be confirmed by using (3.13).

Finally, (3.14), (3.15) and (3.16) assert that $\underline{w} (\cdot, t) \in C^{2} ([0, R^{n}] ∖ N (t))$ for all $t > 0$ and that ${\underline{w}}_{s s} (s, t) ⩽ 0$ for all $t > 0$ and $s \in (0, R^{n}) ∖ N (t)$ , so that (3.11) becomes a consequence of (3.16) and (3.10). □

The motivation for the choice of the structure in (3.4), and especially for the selection of the lower order contributions $b e^{ρ t}$ to the above definition of ${(y^{(a, b)})}_{a > a_{0}, b > 0}$ , originates from the following basic observation on a certain subsolution property thereby asserted within suitable regions of the $(a, b)$ plane. Lemma 3.3 will be relied on already in our analysis of ${\underline{w}}_{in}$ in Lemma 4.1, but beyond this to a yet more substantial extent in our considerations concerned with ${\underline{w}}_{mid}$ detailed in Lemma 4.2. Lemma 3.3.
Let $n ⩾ 2$ , $m \in (- \infty, 1 - \frac{2}{n})$ and $χ > 0$ , and let $k ⩾ 0$ , $l ⩾ 0$ , $L > 0$ , $λ \in (1, 2]$ as well as $a > 0$ and $b > 0$ be such that with γ, δ, κ and ρ taken from Lemma 3.1 we have $\begin{matrix} (3.17) & b \cdot (χ - ρ) ⩾ k χ {(a + b)}^{\frac{1}{γ}} + l {(a + b)}^{1 - (1 - γ + κ γ) (λ - 1)} + L {(a + b)}^{1 - δ} . \end{matrix}$ Then the function $y = y^{(a, b)}$ defined in ( 3.4 ) satisfies $\begin{matrix} (3.18) & y^{'} (t) + L y^{1 - δ} (t) ⩽ (y (t) - k y^{\frac{1}{γ}} (t)) \cdot (χ - \frac{l}{y^{(1 - γ + κ γ) (λ - 1)} (t)}) for all t > 0 . \end{matrix}$
Proof.
Since $e^{\frac{χ}{γ} t} ⩽ e^{ρ t}$ and $e^{[1 - (1 - γ + κ γ) (λ - 1)] \cdot χ t} ⩽ e^{ρ t}$ for all $t > 0$ by (3.3), we can estimate $\begin{array}{l} (y (t) - k y^{\frac{1}{γ}} (t)) \cdot (χ - \frac{l}{y^{(1 - γ + κ γ) (λ - 1)} (t)}) \\ = χ y (t) - k χ y^{\frac{1}{γ}} (t) - l y^{1 - (1 - γ + κ γ) (λ - 1)} (t) + k l y^{\frac{1}{γ} - (1 - γ + κ γ) (λ - 1)} (t) \\ ⩾ χ \cdot (a e^{χ t} + b e^{ρ t}) - k χ \cdot {(a e^{χ t} + b e^{ρ t})}^{\frac{1}{γ}} - l \cdot {(a e^{χ t} + b e^{ρ t})}^{1 - (1 - γ + κ γ) (λ - 1)} \\ ⩾ a χ e^{χ t} + b χ e^{ρ t} - k χ {(a + b)}^{\frac{1}{γ}} e^{\frac{χ}{γ} t} - l {(a + b)}^{1 - (1 - γ + κ γ) (λ - 1)} e^{[1 - (1 - γ + κ γ) (λ - 1)] \cdot χ t} \\ ⩾ a χ e^{χ t} + b χ e^{ρ t} - k χ {(a + b)}^{\frac{1}{γ}} e^{ρ t} - l {(a + b)}^{1 - (1 - γ + κ γ) (λ - 1)} e^{ρ t} \\ = a χ e^{χ t} - {k χ {(a + b)}^{\frac{1}{γ}} + l {(a + b)}^{1 - (1 - γ + κ γ) (λ - 1)} - b χ} \cdot e^{ρ t} \\ ⩾ a χ e^{χ t} + b ρ e^{ρ t} + L {(a + b)}^{1 - δ} e^{ρ t} for all t > 0 \end{array}$ according to (3.17). As, on the other hand, (3.3) also ensures that $e^{(1 - δ) χ t} ⩽ e^{ρ t}$ for all $t > 0$ and hence $\begin{array}{rcl} y^{'} (t) + L y^{1 - δ} (t) & = & a χ e^{χ t} + b ρ e^{ρ t} + L \cdot {(a e^{χ t} + b e^{ρ t})}^{1 - δ} \\ ⩽ & a χ e^{χ t} + b ρ e^{ρ t} + L {(a + b)}^{1 - δ} e^{(1 - δ) χ t} \\ ⩽ & a χ e^{χ t} + b ρ e^{ρ t} + L {(a + b)}^{1 - δ} e^{ρ t} for all t > 0, \end{array}$ this already yields (3.18). □

4. Parameter conditions ensuring $P \underline{w} ⩽ 0$

Within the time-dependent part of the three subdomains of $[0, R^{n}] \times [0, \infty)$ in (3.9) which touches the spatial origin, according to the spatially linear behavior of the functions in (3.9) and (3.6) the following derivation of a subsolution feature may fully ignore the diffusive part in (2.6). As a consequence, we will here rely on Lemma 3.3 only in a comparatively weak form in which the seond summand on the right of (3.18) is actually absent.

Lemma 4.1.
Let $n ⩾ 2$ and $R > 0$ , suppose that ( 1.6 ), ( 1.7 ), ( 1.9 ) and ( 1.10 ) hold with some $m \in (- \infty, 1 - \frac{2}{n})$ , $K_{D} > 0$ , $χ > 0$ , $λ \in (1, 2]$ , $K_{S} > 0$ and $ξ_{0} > 0$ , and let γ, δ, ρ and $a_{0}$ be as in Lemma 3.1 and Lemma 3.2 . Then one can fix $μ_{in} > 0$ in such a way that for all $μ > 0$ there exists $a_{in} = a_{in}^{(μ)} ⩾ a_{0}$ such that for each $a > a_{in}$ one can find $b_{in} = b_{in}^{(μ, a)} > 0$ with the property that for any $b > b_{in}$ , with $y = y^{(μ, a, b)}$ and $P$ taken from ( 3.4 ) and ( 2.6 ) we have $\begin{matrix} (4.1) & (P {\underline{w}}_{in}) (s, t) ⩽ 0 for all t > 0 and s \in (0, \frac{1}{y (t)}) . \end{matrix}$
Proof.
We let $θ_{in} : = \frac{ξ_{0}}{n}$ and $\begin{matrix} (4.2) & μ_{in} : = \frac{n}{R^{n}} \cdot (\frac{γ}{γ - 1} + R^{n}) \cdot θ_{in}, \end{matrix}$ and pick any $μ > μ_{in}$ , We then take $θ_{\infty} = θ_{\infty}^{(μ)}$ as in (3.12), and set $k \equiv k^{(μ)} : = \frac{μ}{n θ_{\infty}}$ , $l \equiv l^{(μ)} : = \frac{K_{S}}{n^{λ - 1} θ_{\infty}^{λ - 1}}$ and $\begin{matrix} (4.3) & a_{in} \equiv a_{in}^{(μ)} : = max {a_{0}, {(\frac{l}{χ})}^{\frac{1}{λ - 1}}, k}, \end{matrix}$ and given $a > a_{in}$ we pick $b_{in} = b_{in}^{(μ, a)} > 0$ large enough such that $\begin{matrix} (4.4) & b \cdot (χ - ρ) ⩾ k χ + l {(a + b)}^{2 - λ} for all b ⩾ b_{in} . \end{matrix}$ Fixing any $b > b_{in}$ and letting $y = y^{(a, b)}$ and ${\underline{w}}_{in} = {\underline{w}}_{in}^{(μ, a, b)}$ be as accordingly obtained from (3.4), (3.5) and (3.6), we can then rely on (4.2) to see that $n θ_{\infty} > ξ_{0}$ and that thus, by (3.11) and (2.2), $\begin{array}{rcl} n \partial_{s} {\underline{w}}_{in} (s, t) > ξ_{0} for all t > 0 and s \in (0, \frac{1}{y (t)}) . \end{array}$ Therefore, (1.10) applies so as to assert that thanks to (3.14) and the fact that $θ (t) ⩾ θ_{\infty}$ for all $t > 0$ by (3.10), $\begin{array}{rcl} S (n \partial_{s} {\underline{w}}_{in} (s, t)) \cdot n \partial_{s} {\underline{w}}_{in} (s, t) & ⩾ & {\frac{χ}{n \partial_{s} {\underline{w}}_{in} (s, t)} - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{in} (s, t))}^{λ}}} \cdot n \partial_{s} {\underline{w}}_{in} (s, t) \\ = & χ - \frac{K_{S}}{n^{λ - 1} θ^{λ - 1} (t) y^{λ - 1} (t)} \\ ⩾ & χ - \frac{K_{S}}{n^{λ - 1} θ_{\infty}^{λ - 1} y^{λ - 1} (t)} \\ = & χ - \frac{l}{y^{λ - 1} (t)} for all t > 0 and s \in (0, \frac{1}{y (t)}) . \end{array}$ Since here the second lower bound for a implied by (4.3) warrants that $\begin{array}{rcl} χ - \frac{l}{y^{λ - 1} (t)} ⩾ χ - \frac{l}{a^{λ - 1}} > 0 for all t > 0, \end{array}$ and since moreover $\begin{array}{rcl} y (t) - \frac{μ}{n θ (t)} ⩾ a - \frac{μ}{n θ_{\infty}} = a - k > 0 for all t > 0 \end{array}$ due to the third requirement expressed by (4.3), we can estimate $\begin{array}{l} ({\underline{w}}_{in} (s, t) - \frac{μ}{n} s) \cdot S (n \partial_{s} {\underline{w}}_{in} (s, t)) \cdot n \partial_{s} {\underline{w}}_{in} (s, t) \\ = θ (t) s \cdot (y (t) - \frac{μ}{n θ (t)}) \cdot S (n \partial_{s} {\underline{w}}_{in} (s, t)) \cdot n \partial_{s} {\underline{w}}_{in} (s, t) \\ ⩾ θ (t) s \cdot (y (t) - k) \cdot (χ - \frac{l}{y^{λ - 1} (t)}) for all t > 0 and s \in (0, \frac{1}{y (t)}) . \end{array}$ In view of (2.6) and again (3.14), we thus obtain that due to the inequality $θ^{'} ⩽ 0$ , $\begin{array}{l} \begin{matrix} (P {\underline{w}}_{in}) (s, t) = & θ^{'} (t) y (t) s + θ (t) y^{'} (t) s \\ - ({\underline{w}}_{in} (s, t) - \frac{μ}{n} s) \cdot S (n \partial_{s} {\underline{w}}_{in} (s, t)) \cdot n \partial_{s} {\underline{w}}_{in} (s, t) \\ ⩽ & θ (t) y^{'} (t) s - θ (t) s \cdot (y (t) - k) \cdot (χ - \frac{l}{y^{λ - 1} (t)}) \\ = & θ (t) s \cdot {y^{'} (t) - (y (t) - k) \cdot (χ - \frac{l}{y^{λ - 1} (t)})} \end{matrix} \\ (4.5) & for all t > 0 and s \in (0, \frac{1}{y (t)}) . \end{array}$ Since (4.4) warrants validity of (3.17) with $L : = 0$ , upon an application of Lemma 3.3, trivially estimating $y^{\frac{1}{γ}} ⩾ 1$ , we obtain (4.1) from (4.5). □

Within the transition region appearing in (3.9), our analysis now makes full use of Lemma 3.3 in order to estimate the influence of the diffusive contribution to the operator $P$ in terms of the corresponding taxis-related expressions. Through Lemma 3.1 relying on the subcriticality requirement $m < 1 - \frac{2}{n}$ in its full strength, namely, the following argument is based on the observation that this dissipative action can be seen to play the role of a lower-power nonlinearity overbalanced by the respective transport mechanism, at its core quantifiable by including the sublinear summand on the right of (3.18).
Lemma 4.2.
Let $n ⩾ 2$ and $R > 0$ , let ( 1.6 ), ( 1.7 ), ( 1.9 ) and ( 1.10 ) be fulfilled with some $m \in (- \infty, 1 - \frac{2}{n})$ , $K_{D} > 0$ , $χ > 0$ , $λ \in (1, 2]$ , $K_{S} > 0$ and $ξ_{0} > 0$ , and let γ, δ, ρ and $a_{0}$ be as introduced in Lemma 3.1 and Lemma 3.2 . Then there exists $μ_{mid} > 0$ such that for all $μ > μ_{mid}$ one can fix $a_{mid} = a_{mid}^{(μ)} ⩾ a_{0}$ such that for any choice of $a > a_{mid}$ , it is possible to pick $b_{mid} = b_{mid}^{(μ, a)} > 0$ such that for arbitrary $b > b_{mid}$ , the function ${\underline{w}}_{mid} = {\underline{w}}_{mid}^{(μ, a, b)}$ from ( 3.7 ), with $y = y^{(a, b)}$ as accordingly defined in ( 3.4 ), satisfies $\begin{matrix} (4.6) & (P {\underline{w}}_{mid}) (s, t) ⩽ 0 for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}) \end{matrix}$ where $P$ is as in ( 2.6 ).
Proof.
We let $\begin{matrix} (4.7) & θ_{mid} : = max {\frac{ξ_{0}}{n}, \frac{1}{n} \cdot {(\frac{2 K_{S}}{χ})}^{\frac{1}{λ - 1}}, {(\frac{8 n^{m + 1} γ K_{D}}{χ})}^{\frac{1}{1 - m}}} \end{matrix}$ as well as $\begin{matrix} (4.8) & μ_{mid} : = \frac{n}{R^{n}} \cdot (\frac{γ}{γ - 1} + R^{n}) \cdot θ_{mid}, \end{matrix}$ and given $μ > μ_{mid}$ we choose $a_{mid} = a_{mid}^{(μ)} ⩾ a_{0}$ large enough such that taking $θ_{\infty}$ and κ as in (3.12) and Lemma 3.1, and abbreviating $\begin{matrix} (4.9) & k \equiv k^{(μ)} : = \frac{μ}{n θ_{\infty}}, l \equiv l^{(μ)} : = \frac{K_{S}}{n^{λ - 1} θ_{\infty}^{λ - 1}} and L \equiv L^{(μ)} : = n^{m + 1} γ K_{D} θ_{\infty}^{m - 1}, \end{matrix}$ we have $\begin{matrix} (4.10) & \frac{k}{a_{mid}^{\frac{γ - 1}{γ}}} ⩽ \frac{1}{2} \end{matrix}$ and $\begin{matrix} (4.11) & a_{mid}^{(1 - κ) (γ - 1)} ⩾ 8 . \end{matrix}$ Given $a > a_{mid}$ , we then pick $b_{mid} = b_{mid}^{(μ, a)} > 0$ in such a way that $\begin{matrix} (4.12) & b \cdot (χ - ρ) ⩾ k χ {(a + b)}^{\frac{1}{γ}} + l {(a + b)}^{1 - (1 - γ + κ γ) (λ - 1)} + L {(a + b)}^{1 - δ} for all b > b_{mid}, \end{matrix}$ and for $μ > μ_{mid}$ , $a > a_{mid}^{(μ)}$ and $b > b_{mid}^{(μ, a)}$ we now take $y = y^{(a, b)}$ , $θ = θ^{(μ, a, b)}$ and ${\underline{w}}_{mid} = {\underline{w}}_{mid}^{(μ, a, b)}$ as correspondingly defined in (3.4), (3.5) and (3.7), first observing that then (4.8) and the first restriction in (4.7) together with (3.11) guarantees that $\begin{matrix} (4.13) & n \partial_{s} {\underline{w}}_{mid} (s, t) ⩾ n θ_{\infty} ⩾ n θ_{mid} ⩾ ξ_{0} for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{\frac{γ - 1}{γ}} (t)}) . \end{matrix}$ Therefore, (1.9) and (1.10) become applicable so as to imply that $\begin{array}{l} - n^{2} s^{2 - \frac{2}{n}} D (n \partial_{s} {\underline{w}}_{mid} (s, t)) \partial_{s}^{2} {\underline{w}}_{mid} (s, t) \\ ⩽ - n^{2} K_{D} s^{2 - \frac{2}{n}} \cdot {(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{m - 1} \partial_{s}^{2} {\underline{w}}_{mid} (s, t) \\ = n^{2} K_{D} s^{2 - \frac{2}{n}} \cdot {n θ (t) y^{1 - γ} (t) s^{- γ}}^{m - 1} \cdot γ θ (t) y^{1 - γ} (t) s^{- γ - 1} \\ (4.14) & = n^{m + 1} γ K_{D} θ^{m} (t) y^{m (1 - γ)} (t) s^{- m γ + 1 - \frac{2}{n}}, \end{array}$ and that $\begin{array}{rcl} (4.15) & S (n \partial_{s} {\underline{w}}_{mid} (s, t)) \cdot n \partial_{s} {\underline{w}}_{mid} (s, t) & ⩾ & χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{λ - 1}} \end{array}$ for all $t > 0$ and $s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)})$ , where since (4.8) and the second condition contained in (4.7) ensure that $n \partial_{s} {\underline{w}}_{mid} (s, t) ⩾ n θ_{\infty} > n θ_{mid} ⩾ {(\frac{2 K_{S}}{χ})}^{\frac{1}{λ - 1}}$ for any such t and s, we particularly obtain that $\begin{matrix} (4.16) & S (n \partial_{s} {\underline{w}}_{mid} (s, t)) \cdot n \partial_{s} {\underline{w}}_{mid} (s, t) ⩾ \frac{χ}{2} for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{\frac{γ - 1}{γ}} (t)}) . \end{matrix}$ When combined with (3.7), the inequality in (3.10) moreover entails that $\begin{array}{rcl} \frac{1}{θ (t)} \cdot {{\underline{w}}_{mid} (s, t) - \frac{μ}{n} s} & = & \frac{γ}{γ - 1} - \frac{1}{γ - 1} y^{1 - γ} (t) s^{1 - γ} - \frac{μ}{n θ (t)} s \\ ⩾ & \frac{γ}{γ - 1} - \frac{1}{γ - 1} y^{1 - γ} (t) \cdot {(\frac{1}{y (t)})}^{1 - γ} - \frac{μ}{n θ_{\infty} y^{\frac{γ - 1}{γ}} (t)} \\ = & 1 - \frac{k}{y^{\frac{γ - 1}{γ}} (t)} for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}), \end{array}$ whence in view of the positivity statement on $S (n \partial_{s} {\underline{w}}_{mid}) \cdot n \partial_{s} {\underline{w}}_{mid}$ implied by (4.16), from (4.14), (4.15), (3.15) and again (3.10) we infer that for all $t > 0$ and $s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)})$ , $\begin{array}{rcl} \frac{1}{θ (t)} \cdot (P {\underline{w}}_{mid}) (s, t) & ⩽ & \frac{γ}{γ - 1} \frac{θ^{'} (t)}{θ (t)} - \frac{1}{γ - 1} \frac{θ^{'} (t)}{θ (t)} y^{1 - γ} (t) s^{1 - γ} + y^{- γ} (t) y^{'} (t) s^{1 - γ} \\ + n^{m + 1} γ K_{D} θ^{m - 1} (t) y^{m (1 - γ)} (t) s^{- m γ + 1 - \frac{2}{n}} \\ - (1 - \frac{k}{y^{\frac{γ - 1}{γ}} (t)}) \cdot (χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{λ - 1}}) \\ ⩽ & \frac{γ}{γ - 1} \frac{θ^{'} (t)}{θ (t)} - \frac{1}{γ - 1} \frac{θ^{'} (t)}{θ (t)} y^{1 - γ} (t) s^{1 - γ} + y^{- γ} (t) y^{'} (t) s^{1 - γ} \\ + n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{m (1 - γ)} (t) s^{- m γ + 1 - \frac{2}{n}} \\ (4.17) & - (1 - \frac{k}{y^{\frac{γ - 1}{γ}} (t)}) \cdot (χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{λ - 1}}) . \end{array}$ Once more estimating $y^{1 - γ} (t) s^{1 - γ} ⩽ 1$ for $t > 0$ and $s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)})$ , according to the nonpositivity of $θ^{'}$ implied by (3.13) we see that here $\begin{array}{rcl} \frac{γ}{γ - 1} \frac{θ^{'} (t)}{θ (t)} - \frac{1}{γ - 1} \frac{θ^{'} (t)}{θ (t)} y^{1 - γ} (t) s^{1 - γ} ⩽ \frac{θ^{'} (t)}{θ (t)} ⩽ 0 for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}) \end{array}$ so that within the entire region under consideration we obtain from (4.17) that $\begin{array}{l} \begin{matrix} \frac{y (t)}{θ (t)} \cdot (P {\underline{w}}_{mid}) (s, t) ⩽ & y^{1 - γ} (t) y^{'} (t) s^{1 - γ} \\ + n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{1 + m (1 - γ)} (t) s^{- m γ + 1 - \frac{2}{n}} \\ - (y (t) - k y^{\frac{1}{γ}} (t)) \cdot (χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{λ - 1}}) \end{matrix} \\ (4.18) & for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}) . \end{array}$ To derive (4.6) from this, we now use that $y (t) ⩾ a > 1$ for all $t > 0$ , and that the number κ from Lemma 3.1 satisfies $\frac{γ - 1}{γ} < κ < 1$ , in splitting $(\frac{1}{y (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}) = (\frac{1}{y (t)}, \frac{1}{y^{κ} (t)}] \cup (\frac{1}{y^{κ} (t)}, \frac{1}{y^{(γ - 1) / γ} (t)})$ for $t > 0$ , where for numbers s from the former range we continue to estimate $s ⩾ \frac{1}{y (t)}$ to find that $\begin{matrix} (4.19) & y^{1 - γ} (t) y^{'} (t) s^{1 - γ} ⩽ y^{'} (t) for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{κ} (t)}], \end{matrix}$ while relying on the positivity of $- m γ + 1 - \frac{2}{n}$ , as asserted by Lemma 3.1, we may use that for any such $(s, t)$ we have $s ⩽ \frac{1}{y^{κ} (t)}$ in deriving the inequality $\begin{array}{rcl} n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{1 + m (1 - γ)} (t) s^{- m γ + 1 - \frac{2}{n}} & ⩽ & n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{1 + m (1 - γ) - κ \cdot (- m γ + 1 - \frac{2}{n})} (t) \\ (4.20) & ⩽ & L y^{1 - δ} (t) for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{κ} (t)}], \end{array}$ because $1 + m (1 - γ) - κ \cdot (- m γ + 1 - \frac{2}{n}) ⩽ 1 - δ$ by (3.2), and again because $y (t) > 1$ for all $t > 0$ . Apart from that, again making use of the inequality $s ⩽ \frac{1}{y^{κ} (t)}$ we can control the cross-diffusive contribution to (4.18) from below in this domain upon observing that by (3.15), (3.10) and (4.9), $\begin{array}{rcl} χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{λ - 1}} & = & χ - \frac{K_{S}}{{(n θ (t) y^{1 - γ} (t) s^{- γ})}^{λ - 1}} \\ ⩾ & χ - \frac{K_{S}}{{(n θ_{\infty} y^{1 - γ} (t) y^{κ γ} (t))}^{λ - 1}} \\ = & χ - \frac{l}{y^{(1 - γ + κ γ) (λ - 1)} (t)} for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{κ} (t)}] . \end{array}$ As (4.12) warrants validity of (3.17), from (4.18), (4.19), (4.20), Lemma 3.3 and (3.4), namely, we therefore conclude that $\begin{array}{rcl} \frac{y (t)}{θ (t)} \cdot (P {\underline{w}}_{mid}) (s, t) & ⩽ & y^{'} (t) + L y^{1 - δ} (t) - (y (t) - k y^{\frac{1}{γ}} (t)) \cdot (χ - \frac{l}{y^{(1 - γ + κ γ) (λ - 1)} (t)}) \\ (4.21) & ⩽ & 0 for all t > 0 and s \in (\frac{1}{y (t)}, \frac{1}{y^{κ} (t)}] . \end{array}$ In the corresponding complementary region, however, we may utilize (4.10) along with the inequality $y ⩾ a ⩾ a_{mid}$ to see that $\begin{array}{rcl} \frac{k}{y^{\frac{γ - 1}{γ}} (t)} ⩽ \frac{k}{a_{mid}^{\frac{γ - 1}{γ}}} ⩽ \frac{1}{2} for all t > 0, \end{array}$ so that by (4.16), $\begin{matrix} (4.22) & (y (t) - k y^{\frac{1}{γ}} (t)) \cdot (χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{mid} (s, t))}^{λ - 1}}) ⩾ \frac{χ}{4} y (t) for all t > 0 . \end{matrix}$ On the other hand, we may now use that within this range of variables we have $s > \frac{1}{y^{κ} (t)}$ in verifying that since clearly $y^{'} (t) ⩽ χ y (t)$ for all $t > 0$ by (3.4) and the inequality $ρ ⩽ χ$ , $\begin{array}{rcl} y^{1 - γ} (t) y^{'} (t) s^{1 - γ} & ⩽ & y^{- (1 - κ) (γ - 1)} (t) y^{'} (t) ⩽ χ y^{1 - (1 - κ) (γ - 1)} (t) \\ = & \frac{χ}{y^{(1 - κ) (γ - 1)} (t)} \cdot y (t) ⩽ \frac{χ}{a^{(1 - κ) (γ - 1)}} \cdot y (t) \\ (4.23) & ⩽ & \frac{χ}{8} \cdot y (t) for all t > 0 and s \in (\frac{1}{y^{κ} (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}) \end{array}$ due to (4.11), whereas drawing on the inequality $s ⩽ \frac{1}{y^{\frac{γ - 1}{γ}} (t)}$ for such $(s, t)$ we see that $\begin{array}{l} n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{1 + m (1 - γ)} (t) s^{- m γ + 1 - \frac{2}{n}} \\ ⩽ n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{1 + m (1 - γ) - \frac{γ - 1}{γ} \cdot (- m γ + 1 - \frac{2}{n})} (t) \\ = n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y^{- \frac{γ - 1}{γ} \cdot (1 - \frac{2}{n})} (t) \cdot y (t) \\ (4.24) & ⩽ n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y (t) for all t > 0 and s \in (\frac{1}{y^{κ} (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}), \end{array}$ because $n ⩾ 2$ and $y ⩾ 1$ .

A combination of (4.22) with (4.23), (4.24) and (4.18) hence shows that within this outer part of the intermediate region we have $\begin{array}{l} \begin{matrix} \frac{y (t)}{θ (t)} \cdot (P {\underline{w}}_{mid}) (s, t) & ⩽ \frac{χ}{8} y (t) + n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} y (t) - \frac{χ}{4} y (t) \\ = (- \frac{χ}{8} + n^{m + 1} γ K_{D} θ_{\infty}^{m - 1}) \cdot y (t) \end{matrix} \\ (4.25) & for all t > 0 and s \in (\frac{1}{y^{κ} (t)}, \frac{1}{y^{(γ - 1) / γ} (t)}) . \end{array}$ It thus remains to observe that our selection in (4.8) ensures that since $m - 1 ⩽ 0$ , $\begin{array}{rcl} n^{m + 1} γ K_{D} θ_{\infty}^{m - 1} & = & n^{m + 1} γ K_{D} \cdot {\frac{\frac{μ R^{n}}{n}}{\frac{γ}{γ - 1} + R^{n}}}^{m - 1} \\ ⩽ & n^{m + 1} γ K_{D} \cdot {\frac{\frac{μ_{mid} R^{n}}{n}}{\frac{γ}{γ - 1} + R^{n}}}^{m - 1} \\ = & \frac{χ}{8} . \end{array}$ Therefore, namely, (4.25) together with (4.21) establishes (4.6). □

In the remaining outer region from (3.9), again due to spatial constancy of ${\underline{w}}_{s}$ we may treat $P$ as an essentially hyperbolic operator. Lemma 4.3.
Let $n ⩾ 2$ and $R > 0$ , and assume ( 1.6 ), ( 1.7 ), ( 1.9 ) and ( 1.10 ) with some $m \in (- \infty, 1 - \frac{2}{n})$ , $K_{D} > 0$ , $χ > 0$ , $λ \in (1, 2]$ , $K_{S} > 0$ and $ξ_{0} > 0$ . Then letting γ, δ, ρ and $a_{0}$ be as in Lemma 3.1 and Lemma 3.2 , one can find $μ_{out} > 0$ with the property that whenever $μ > μ_{out}$ , there exists $a_{out} = a_{out}^{(μ)} ⩾ a_{0}$ such that for any choice of $a > a_{out}$ and $b > 0$ , the function ${\underline{w}}_{out} = {\underline{w}}_{out}^{(μ, a, b)}$ , as defined in ( 3.8 ) with $y = y^{(a, b)}$ and $θ = θ^{(μ, a, b)}$ given by ( 3.4 ) and ( 3.5 ), satisfies $\begin{matrix} (4.26) & (P {\underline{w}}_{out}) (s, t) ⩽ 0 for all t > 0 and s \in (\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n}), \end{matrix}$ where $P$ is as in ( 2.6 ).
Proof.
With $\begin{matrix} (4.27) & θ_{out} : = max {{(\frac{2 K_{S}}{n^{λ - 1} χ})}^{\frac{1}{λ - 1}}, \frac{ξ_{0}}{n}}, \end{matrix}$ we let $\begin{matrix} (4.28) & μ_{out} : = \frac{n}{R^{n}} \cdot (\frac{γ}{γ - 1} + R^{n}) \cdot θ_{out}, \end{matrix}$ and given $μ > μ_{out}$ we observe that the number $θ_{\infty} = θ_{\infty}^{(μ)}$ accordingly defined in (3.12) then satisfies $θ_{\infty} > θ_{out}$ due to (4.28) and thus, by (4.27), $\begin{matrix} (4.29) & \frac{K_{S}}{n^{λ - 1} θ_{\infty}^{λ - 1}} ⩽ \frac{χ}{2} . \end{matrix}$ Keeping any such μ fixed henceforth, we next note that $\begin{array}{rcl} φ (σ) \equiv φ^{(μ)} (σ) : = \frac{\frac{χ μ R^{n}}{n} \cdot σ}{{[\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} \cdot σ]}^{2}} - \frac{\frac{χ μ}{2 n} \cdot (1 - σ)}{1 - σ + \frac{γ - 1}{γ} R^{n}}, σ \in [0, 1], \end{array}$ is well-defined and continuous with $\begin{array}{rcl} φ (0) = - \frac{\frac{χ μ}{2 n}}{1 + \frac{γ - 1}{γ} R^{n}} < 0, \end{array}$ so that we can pick $σ_{0} \in (0, 1)$ such that $\begin{matrix} (4.30) & φ (σ) ⩽ 0 for all σ \in (0, σ_{0}), \end{matrix}$ and we thereupon choose $a_{out} = a_{out}^{(μ)} ⩾ a_{0}$ large enough satisfying $\begin{matrix} (4.31) & n a_{out} θ_{\infty} ⩾ ξ_{0} \end{matrix}$ and $\begin{matrix} (4.32) & a_{out}^{- \frac{γ - 1}{γ}} ⩽ σ_{0} . \end{matrix}$ For arbitrary $a > a_{out}$ and $b > 0$ , we then let $y = y^{(a, b)}$ , $θ = θ^{(μ, a, b)}$ and ${\underline{w}}_{out} = {\underline{w}}_{out}^{(μ, a, b)}$ by as in (3.4), (3.5) and (3.8), and once again begin by using (3.11) to see that $n \partial_{s} {\underline{w}}_{out} (s, t) ⩾ ξ_{0}$ for all $t > 0$ and $s \in (\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n})$ by the second restriction in (4.27), and that hence $\begin{array}{rcl} S (n \partial_{s} {\underline{w}}_{out} (s, t)) \cdot n \partial_{s} {\underline{w}}_{out} (s, t) & ⩾ & χ - \frac{K_{S}}{{(n \partial_{s} {\underline{w}}_{out} (s, t))}^{λ - 1}} \\ = & χ - \frac{K_{S}}{n^{λ - 1} θ^{λ - 1} (t)} \\ ⩾ & χ - \frac{K_{S}}{n^{λ - 1} θ_{\infty}^{λ - 1}} \\ (4.33) & ⩾ & \frac{χ}{2} for all t > 0 and s \in (\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n}) \end{array}$ thanks to (3.10) and (4.29). Apart from that, using that $(0, 1) ∋ σ \mapsto \frac{1 - σ}{1 - σ + \frac{γ - 1}{γ} R^{n}}$ is nonincreasing and that $y (t) ⩾ a$ for all $t > 0$ we can estimate $\begin{array}{l} \begin{matrix} {\underline{w}}_{out} (s, t) - \frac{μ}{n} s & = \frac{μ R^{n}}{n} - θ (t) \cdot (R^{n} - s) - \frac{μ}{n} s \\ = \frac{μ}{n} \cdot {1 - \frac{n}{μ} θ (t)} \cdot (R^{n} - s) \\ = \frac{μ}{n} \cdot {1 - \frac{R^{n}}{\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ - 1}{γ}} (t)}} \cdot (R^{n} - s) \\ = \frac{μ}{n} \cdot \frac{\frac{γ}{γ - 1} \cdot (1 - y^{- \frac{γ - 1}{γ}} (t))}{\frac{γ}{γ - 1} \cdot (1 - y^{- \frac{γ - 1}{γ}} (t)) + R^{n}} \cdot (R^{n} - s) \\ = \frac{μ}{n} \cdot \frac{1 - y^{- \frac{γ - 1}{γ}} (t)}{1 - y^{- \frac{γ - 1}{γ}} (t) + \frac{γ - 1}{γ} R^{n}} \cdot (R^{n} - s) \\ ⩾ \frac{μ}{n} \cdot \frac{1 - a^{- \frac{γ - 1}{γ}}}{1 - a^{- \frac{γ - 1}{γ}} + \frac{γ - 1}{γ} R^{n}} \cdot (R^{n} - s) \end{matrix} \\ (4.34) & for all t > 0 and s \in (\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n}), \end{array}$ while combining (3.16) with the rough inequality $y^{'} (t) ⩽ χ y (t)$ for $t > 0$ we see that again since $y (t) ⩾ a$ for all $t > 0$ , $\begin{array}{l} \begin{matrix} \partial_{t} {\underline{w}}_{out} (s, t) & = - θ^{'} (t) \cdot (R^{n} - s) \\ = \frac{\frac{μ R^{n}}{n}}{{[\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ - 1}{γ}} (t)]}^{2}} \cdot y^{\frac{1}{γ} - 2} (t) y^{'} (t) \cdot (R^{n} - s) \\ ⩽ \frac{\frac{χ μ R^{n}}{n}}{{[\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} y^{- \frac{γ - 1}{γ}} (t)]}^{2}} \cdot y^{- \frac{γ - 1}{γ}} (t) \cdot (R^{n} - s) \\ ⩽ \frac{\frac{χ μ R^{n}}{n} \cdot a^{- \frac{γ - 1}{γ}}}{{[\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} a^{- \frac{γ - 1}{γ}}]}^{2}} \cdot (R^{n} - s) \end{matrix} \\ for all t > 0 and s \in (\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n}) . \end{array}$ Together with (4.33), (4.34) and the fact that $\partial_{s}^{2} {\underline{w}}_{out} (s, t) = 0$ for all $t > 0$ and $s \in (\frac{1}{y^{\frac{γ - 1}{γ}} (t), R^{n}})$ , in line with (2.6) this shows that $\begin{array}{rcl} (P {\underline{w}}_{out}) (s, t) & = & \partial_{t} {\underline{w}}_{out} (s, t) - ({\underline{w}}_{out} (s, t) - \frac{μ}{n} s) \cdot S (n \partial_{s} {\underline{w}}_{out} (s, t)) \cdot n \partial_{s} {\underline{w}}_{out} (s, t) \\ ⩽ & {\frac{\frac{χ μ R^{n}}{n} \cdot a^{- \frac{γ - 1}{γ}}}{{[\frac{γ}{γ - 1} + R^{n} - \frac{γ}{γ - 1} a^{- \frac{γ - 1}{γ}}]}^{2}} - \frac{μ}{n} \cdot \frac{1 - a^{- \frac{γ - 1}{γ}}}{1 - a^{- \frac{γ - 1}{γ}} + \frac{γ - 1}{γ} R^{n}} \cdot \frac{χ}{2}} \cdot (R^{n} - s) \\ = & φ (a^{- \frac{γ - 1}{γ}}) \cdot (R^{n} - s) for all t > 0 and s \in (\frac{1}{y^{(γ - 1) / γ} (t)}, R^{n}) . \end{array}$ Since $a^{- \frac{γ - 1}{γ}} < σ_{0}$ by (4.32), in light of (4.30) this confirms (4.26). □

5. Exponential grow-up. Proof of Theorem 1.1 and Proposition 1.2

Our main result on exponential lower bounds for grow-up of some solutions can now be achieved by combining Lemma 4.1 and Lemma 4.2 with Lemma 4.3, and applying the comparison principle from Lemma 2.3 to an accordingly obtained fixed function of the form in (3.9), and to any solution w determined through (2.2) upon choosing suitably concentrated but smooth initial data in (1.1):

Proof of Theorem 1.1.

We let $μ_{i n}$ , $μ_{m i d}$ and $μ_{out}$ be as given by Lemma 4.1, Lemma 4.2 and Lemma 4.3, and define $μ_{⋆} : = max {μ_{in}, μ_{mid}, μ_{out}}$ . For arbitrary $μ > μ_{⋆}$ , we then take $a_{in}^{(μ)}$ , $a_{mid}^{(μ)}$ and $a_{out}^{(μ)}$ be as correspondingly provided by Lemma 4.1, Lemma 4.2 and Lemma 4.3, and fix any $a > max {a_{in}^{(μ)}, a_{mid}^{(μ)}, a_{out}^{(μ)}}$ . Finally choosing a number $b > max {b_{in}^{(μ, a)}, b_{mid}^{(μ, a)}}$ , with $b_{in}^{(μ, a)}$ and $b_{mid}^{(μ, a)}$ as found in Lemma 4.1 and Lemma 4.2, we let $y = y^{(a, b)}$ , $θ = θ^{(μ, a, b)}$ and $\underline{w} = {\underline{w}}^{(μ, a, b)}$ be as introduced in (3.4), (3.5) and (3.9), and pick any function $w_{0} \in C^{\infty} ([0, R^{n}])$ such that $w_{0 s} > 0$ in $[0, R^{n}]$ and $supp w_{0 s s} \subset (0, R^{n}]$ as well as $\begin{matrix} (5.1) & w_{0} (0) = 0, w_{0} (R^{n}) = \frac{μ R^{n}}{n} and w_{0} (s) > \underline{w} (s, 0) for all s \in (0, R^{n}) . \end{matrix}$ Letting $\begin{array}{rcl} u_{0} (x) : = n w_{0 s} (| x |^{\frac{1}{n}}), x \in \overline{Ω}, \end{array}$ then can readily be seen to define a radially symmetric function from $C^{\infty} (\overline{Ω})$ which is positive in $\overline{Ω}$ and satisfies $\frac{1}{| Ω |} \int_{Ω} u_{0} = μ$ . In light of Lemma 2.1, our hypotheses in (1.7) now guarantee the existence of a global classical solution $(u, v)$ fulfilling (1.11) as well as $u > 0$ in $\overline{Ω} \times [0, \infty)$ , and Lemma 2.2 asserts that the function w defined through (2.2) satisfies $(P w) (s, t) = 0$ for all $(s, t) \in (0, R^{n}) \times (0, \infty)$ , with $P$ taken from (2.6). But since for all $t > 0$ , the function $\underline{w} (\cdot, t)$ belongs to $W^{2, \infty} ((0, R^{n})) \cap C^{2} ([0, R^{n}] ∖ N (t))$ , with $N (t) : = {\frac{1}{y (t)}, \frac{1}{y^{\frac{γ - 1}{γ}} (t)}}$ , and satisfies $(P \underline{w}) (s, t) ⩽ 0$ for all $t > 0$ and any $s \in (0, R^{n}) ∖ N (t)$ according to Lemma 4.1, Lemma 4.2, Lemma 4.3 and our selections of μ, a and b, we may employ Lemma 2.3 to infer from (5.1) that $w (s, t) ⩾ \underline{w} (s, t)$ for all $s \in [0, R^{n}]$ and $t ⩾ 0$ . In particular, due to the mean value theorem this means that $\begin{array}{rcl} {‖ w_{s} (\cdot, t) ‖}_{L^{\infty} ((0, R^{n}))} & ⩾ & \frac{w (\frac{1}{y (t)}, t) - w (0, t)}{\frac{1}{y (t)}} \\ = & y (t) w (\frac{1}{y (t)}, t) \\ ⩾ & y (t) \underline{w} (\frac{1}{y (t)}, t) \\ = & y (t) \cdot {θ (t) y (t) \cdot \frac{1}{y (t)}} \\ = & θ (t) y (t) \\ ⩾ & θ_{\infty} y (t) for all t ⩾ 0 \end{array}$ according to (3.9), (3.6) and (3.10), where again $θ_{\infty}$ is as in (3.12). But since $y (t) ⩾ a e^{χ t}$ for all $t ⩾ 0$ by (3.4), in view of (2.3) this entails that $\begin{array}{rcl} {‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} = n ‖ w_{s} ‖_{L^{\infty} ((0, R^{n}))} ⩾ n a θ_{\infty} e^{χ t} for all t ⩾ 0 \end{array}$ and hence establishes (1.12). □

The complementary statement on exponential upper bounds, finally, will result from a comparison argument applied to the original problem:

Proof of Proposition 1.2.

Abbreviating $z_{0} : = max {ξ_{0}, ‖ u_{0} ‖_{L^{\infty} (Ω)}}$ , we compute the solution of the Bernoulli-type initial value problem $\begin{matrix} (5.2) & \{\begin{matrix} z^{'} (t) = χ z (t) + K_{S} z^{2 - λ} (t), t > 0, \\ z (0) = z_{0}, \end{matrix} \end{matrix}$ according to $\begin{matrix} (5.3) & z (t) = {(z_{0}^{λ - 1} + \frac{K_{S}}{χ}) e^{(λ - 1) χ t} - \frac{K_{S}}{χ}}^{\frac{1}{λ - 1}}, t ⩾ 0, \end{matrix}$ and observe that $z (0) ⩾ u_{0} (x)$ for all $x \in Ω$ , that $z (t) ⩾ z_{0} ⩾ ‖ u_{0} ‖_{L^{\infty} (Ω)} ⩾ μ$ for all $t > 0$ , and that moreover $z (t) ⩾ z_{0} ⩾ ξ_{0}$ for all $t > 0$ . Consequently, for $\begin{array}{rcl} \overline{u} (x, t) : = z (t), x \in \overline{Ω}, t ⩾ 0, \end{array}$ we have $\overline{u} (x, 0) ⩾ u_{0} (x)$ for all $x \in Ω$ as well as $\overline{u} (x, t) ⩾ μ$ for all $(x, t) \in Ω \times (0, \infty)$ . Since furthermore, thanks to (1.13), $S (\overline{u}) ⩽ \frac{χ}{\overline{u}} + \frac{K_{S}}{{\overline{u}}^{λ}}$ in $Ω \times (0, \infty)$ , we obtain that $\begin{array}{l} {\overline{u}}_{t} - \nabla \cdot (D (u) \nabla \overline{u}) + (S (u) + u S^{'} (u)) \nabla v \cdot \nabla \overline{u} - (\overline{u} - μ) \overline{u} S (\overline{u}) \\ = {\overline{u}}_{t} - (\overline{u} - μ) \overline{u} S (\overline{u}) \\ ⩾ {\overline{u}}_{t} - (\overline{u} - μ) \overline{u} \cdot (\frac{χ}{\overline{u}} + \frac{K_{S}}{{\overline{u}}^{λ}}) \\ = {\overline{u}}_{t} - χ \overline{u} - K_{S} {\overline{u}}^{2 - λ} + μ \cdot (χ + \frac{K_{S}}{{\overline{u}}^{λ - 1}}) \\ ⩾ {\overline{u}}_{t} - χ \overline{u} - K_{S} {\overline{u}}^{2 - λ} \\ = 0 in Ω \times (0, \infty) \end{array}$ by (5.2). As, on the other hand, (1.1) implies that $\begin{array}{rcl} u_{t} - \nabla \cdot (D (u) \nabla u) + (S (u) + u S^{'} (u)) \nabla v \cdot \nabla u - (u - μ) u S (u) = 0 in Ω \times (0, \infty), \end{array}$ a comparison argument shows that $u ⩽ \overline{u}$ in $\overline{Ω} \times [0, \infty)$ and hence, in particular, $\begin{array}{rcl} {‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} ⩽ z (t) ⩽ {(z_{0}^{λ - 1} + \frac{K_{S}}{χ})}^{\frac{1}{λ - 1}} e^{χ t} \end{array}$ for all $t ⩾ 0$ . □

Footnotes

Acknowledgement

The author acknowledges support of the Deutsche Forschungsgemeinschaft (Project No. 462888149).

References

Blanchet,

J.A.

Carrillo and

Masmoudi, Infinite time aggregation for the critical Patlak-Keller–Segel model in

R^{2}

, Comm. Pure Appl. Math. 61 (2008), 1449–1481. doi:10.1002/cpa.20225.

Cieślak and

Ph.

Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, CR Math. Acad. Sci. Paris 347 (2009), 237–242. doi:10.1016/j.crma.2009.01.016.

Cieślak and

Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller–Segel system in higher dimensions, J. Differ. Eq. 252(10) (2012), 5832–5851. doi:10.1016/j.jde.2012.01.045.

Cieślak and

Stinner, New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models, J. Differ. Eq. 258(6) (2015), 2080–2113. doi:10.1016/j.jde.2014.12.004.

Cieślak and

Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008), 1057–1076. doi:10.1088/0951-7715/21/5/009.

Djie and

Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Analysis: Theory, Methods and Applications 72(2) (2010), 1044–1064. doi:10.1016/j.na.2009.07.045.

Freitag, Blow-up profiles and refined extensibility criteria in quasilinear Keller–Segel systems, J. Math. Anal. Appl. 463 (2018), 964–988. doi:10.1016/j.jmaa.2018.03.052.

Fuest, Blow-up profiles in quasilinear fully parabolic Keller–Segel systems, Nonlinearity 33 (2020), 2306–2334. doi:10.1088/1361-6544/ab7294.

Fuest, On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems, Appl. Anal., to appear.

10.

Fujie and

Jiang, Comparison methods for a Keller–Segel-type model of pattern formations with density-suppressed motilities, Calc. Var. Partial Differential Equations 60 (2021), 92. doi:10.1007/s00526-021-01943-5.

11.

Giga,

Mizoguchi and

Senba, Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Arch. Ration. Mech. Anal. 201 (2011), 549–573. doi:10.1007/s00205-010-0394-7.

12.

I.A.

Guerra and

M.A.

Peletier, Self-similar blow-up for a diffusion-attraction problem, Nonlinearity 17 (2004), 2137–2162. doi:10.1088/0951-7715/17/6/007.

13.

M.A.

Herrero,

Medina and

J.J.L.

Velázquez, Self-similar blow-up for a reaction-diffusion system, J. Comput. Appl. Math. 97 (1998), 99–119. doi:10.1016/S0377-0427(98)00104-6.

14.

M.A.

Herrero and

J.J.L.

Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583–623. doi:10.1007/BF01445268.

15.

M.A.

Herrero and

J.J.L.

Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore Pisa Cl. Sci. 24 (1997), 633–683.

16.

Hillen and

K.J.

Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183–217. doi:10.1007/s00285-008-0201-3.

17.

Horstmann, The nonsymmetric case of the Keller–Segel model in chemotaxis: Some recent results, Nonlinear Differ. Equ. Appl. 8 (2001), 399–423. doi:10.1007/PL00001455.

18.

Ishida,

Seki and

Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Eq. 256 (2014), 2993–3010. doi:10.1016/j.jde.2014.01.028.

19.

Ishige,

Ph.

Laurençot and

Mizoguchi, Blow-up behavior of solutions to a degenerate parabolic-parabolic Keller–Segel system, Math. Ann. 367 (2017), 461–499. doi:10.1007/s00208-016-1400-7.

20.

Jäger and

Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824. doi:10.1090/S0002-9947-1992-1046835-6.

21.

Kavallaris and

Ph.

Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller–Segel model in a disk, SIAM J. Math. Anal. 40 (2009), 1852–1881. doi:10.1137/080722229.

22.

E.F.

Keller and

L.A.

Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415. doi:10.1016/0022-5193(70)90092-5.

23.

Lankeit, Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller–Segel system, Discr. Cont. Dyn. Syst 13 (2020), 233–255.

24.

M.C.

Lombardo,

Barresi,

Bilotta,

Gargano,

Pantano and

Sammartino, Demyelination patterns in a mathematical model of multiple sclerosis, J. Math. Biol. 75 (2017), 373–417. doi:10.1007/s00285-016-1087-0.

25.

Nagai,

Senba and

Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J. 30 (2000), 463–497.

26.

Naito and

Senba, Self-similar blow-up for a chemotaxis system in higher dimensional domains, in: Mathematical Analysis on the Self-Organization and Self-Similarity, RIMS Kokyuroku Bessatsu, Vol. B15, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, pp. 87–99.

27.

K.J.

Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, J. Theor. Biol. 481 (2019), 162–182. doi:10.1016/j.jtbi.2018.06.019.

28.

Senba, Blowup behavior of radial solutions to Jäger–Luckhaus system in high dimensional domains, Funkcial. Ekvac. 48 (2005), 247–271. doi:10.1619/fesi.48.247.

29.

Senba, A fast blowup solution to an elliptic-parabolic system related to chemotaxis, Adv. Differential Equations 11 (2006), 981–1030.

30.

Ph.

Souplet and

Winkler, Blow-up profiles for the parabolic-elliptic Keller–Segel system in dimensions

n ⩾ 3

, Comm. Math. Phys. 367 (2019), 665–681. doi:10.1007/s00220-018-3238-1.

31.

Stancevic,

C.N.

Angstmann,

J.M.

Murray and

B.I.

Henry, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol. 75 (2013), 774–795. doi:10.1007/s11538-013-9834-5.

32.

Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005.

33.

Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part, J. Math. Pures Appl. 100 (2013), 347–367. doi:10.1016/j.matpur.2013.01.004.

34.

Tao and

Winkler, Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity, J. Differential Eq. 252(1) (2012), 692–715. doi:10.1016/j.jde.2011.08.019.

35.

J.J.L.

Velázquez, Point dynamics for a singular limit of the Keller–Segel model, SIAM J. Appl. Math. 64 (2004), 1198–1223. doi:10.1137/S0036139903433888.

36.

Winkler, Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity, Nonlinearity 30 (2017), 735–764. doi:10.1088/1361-6544/aa565b.

37.

Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities, J. Differential Eq. 266 (2019), 8034–8066. doi:10.1016/j.jde.2018.12.019.

38.

Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller–Segel system, J. Anal. Math. 141 (2020), 585–624. doi:10.1007/s11854-020-0109-4.

39.

Winkler, Complete infinite-time mass aggregation in a quasilinear Keller–Segel system, Preprint.

Exponential grow-up rates in a quasilinear Keller–Segel system

Abstract

Keywords

1. Introduction

Footnotes

Acknowledgement

References