The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption

Abstract

We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the non-perturbed associated ODE. The result in the radial case has been proved in Math. Phys. Anal. Geom. 18(1) (2015), Art. 15. The same approach will be followed here, but the main difference is to construct a Lyapunov functional in similarity variables valid in the non-radial case, which is far from being trivial. That functional is obtained by combining some classical estimates and a new identity of the Pohozaev type obtained by multiplying Eq. (1.7) by $y \cdot \nabla w$ in a suitable weighted space.

Keywords

semilinear wave equation blow-up perturbations conformal exponent Pohozaev identity

1. Introduction

This paper is devoted to the study of blow-up solutions for the following semilinear wave equation: $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t}^{2} u = Δ u + | u |^{p - 1} u + f (u) + g (x, t, \nabla u, \partial_{t} u), \\ u (x, 0) = u_{0} (x) \in H_{loc, u}^{1} (R^{N}) and \partial_{t} u (x, 0) = u_{1} (x) \in L_{loc, u}^{2} (R^{N}), \end{matrix} \end{matrix}$ with conformal power nonlinearity $\begin{matrix} (1.2) & p = p_{c} \equiv 1 + \frac{4}{N - 1}, where N ⩾ 2, \end{matrix}$ and $u (t) : x \in R^{N} \to u (x, t) \in R$ . The space $L_{loc, u}^{2}$ is the set of all $v \in L_{loc}^{2}$ such that $\begin{matrix} ‖ v ‖_{L_{loc, u}^{2}} : = sup_{d \in R^{N}} {(\int_{| x - d | < 1} {| v (x) |}^{2} d x)}^{\frac{1}{2}} < + \infty, \end{matrix}$ and the space $H_{loc, u}^{1} = {v ∣ v, | \nabla v | \in L_{loc, u}^{2}}$ . Moreover, we take $f : R \to R$ and $g : R^{2 N + 2} \to R$ two $C^{1}$ functions satisfying the following conditions: $\begin{matrix} \begin{matrix} (H_{f}) & | f (u) | ⩽ M (1 + \frac{| u |^{p_{c}}}{{log}^{a} (2 + u^{2})}) for all u \in R with (M > 0, a > 1), \\ (H_{g}) & | g (x, t, v, z) | ⩽ M (1 + | v | + | z |) for all x, v \in R^{N}, t, z \in R with (M > 0) . \end{matrix} \end{matrix}$

The Cauchy problem of (1.1) is well posed in $H_{loc, u}^{1} \times L_{loc, u}^{2}$ . This follows from the finite speed of propagation and the well-posedness in $H^{1} \times L^{2}$ , valid whenever $1 < p < p_{S} = 1 + \frac{4}{N - 2}$ . The existence of blow-up solutions $u (t)$ of (1.1) follows from ODE techniques or the energy-based blow-up criterion by Levine [22] (see also [23,35]). More blow-up results can be found in Caffarelli and Friedman [5,6], Kichenassamy and Littman [18,19]. Let us mention the rather surprising result of Killip and Vişan who proved in [21] that the “first” blow-up set ${x_{0} ∣ T (x_{0}) = {min}_{x \in R} T (x)}$ can be any Cantor set. Numerical simulations of blow-up are given by Bizoń et al. (see [1–4]).

In this paper, we will consider $u (x, t)$ a blow-up solution to Eq. (1.1). We define $Γ = {(x, T (x))}$ such that the maximal influence domain $D_{u}$ of u is written as $\begin{matrix} D_{u} = {(x, t) ∣ t < T (x)} . \end{matrix}$ Moreover, from the finite speed of propagation, T is a 1-Lipschitz function. The surface Γ is called the blow-up graph of u. A point $x_{0} \in R^{N}$ is a non-characteristic point if there are $\begin{matrix} (1.3) & δ_{0} = δ_{0} (x_{0}) \in (0, 1) and t_{0} < T (x_{0}) such that u is defined on C_{x_{0}, T (x_{0}), δ_{0}} \cap {t ⩾ t_{0}}, \end{matrix}$ where $C_{\bar{x}, \bar{t}, \bar{δ}} = {(x, t) ∣ t < \bar{t} - \bar{δ} | x - \bar{x} |}$ .

In the case $(f, g) \equiv (0, 0)$ , Eq. (1.1) reduces to the semilinear wave equation: $\begin{matrix} (1.4) & \partial_{t}^{2} u = Δ u + | u |^{p - 1} u, (x, t) \in R^{N} \times [0, T) . \end{matrix}$ It is interesting to recall that when $1 < p ⩽ p_{c}$ , in [24,25] and [26] Merle and Zaag have proved, that if u is a solution of (1.4) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point, then for all $t \in [\frac{3 T (x_{0})}{4}, T (x_{0})]$ , $\begin{array}{rcl} 0 & < & ε_{0} (N, p) ⩽ {(T (x_{0}) - t)}^{\frac{2}{p_{c} - 1}} \frac{‖ u (t) ‖_{L^{2} (B (x_{0}, T (x_{0}) - t))}}{{(T (x_{0}) - t)}^{\frac{N}{2}}} \\ (1.5) & + {(T (x_{0}) - t)}^{\frac{2}{p_{c} - 1} + 1} (\frac{‖ \partial_{t} u (t) ‖_{L^{2} (B (x_{0}, T (x_{0}) - t))}}{{(T (x_{0}) - t)}^{\frac{N}{2}}} + \frac{‖ \nabla u (t) ‖_{L^{2} (B (x_{0}, T (x_{0}) - t))}}{{(T (x_{0}) - t)}^{\frac{N}{2}}}) ⩽ K, \end{array}$ where the constant K depends only on N, p and on an upper bound on $T (x_{0})$ , $1 / T (x_{0})$ , $δ_{0} (x_{0})$ and the initial data in $H_{loc, u}^{1} (R^{N}) \times L_{loc, u}^{2} (R^{N})$ .

Restricting to the one-dimensional case, Merle and Zaag fully described the blow-up dynamics for solutions of (1.4) (see [27,28,30,31]). Later, Côte and Zaag [7] proved the existence of multi-solitons near characteristic points and gave further refinements for general solutions of (1.4), still in one space dimension. Among other results, Merle and Zaag proved that characteristic points are isolated and that the blow-up set ${(x, T (x))}$ is $C^{1}$ near non-characteristic points and corner-shaped near characteristic points. In higher dimensions, the method used in the one-dimensional case does not remain valid because there is no classification of selfsimilar solutions of Eq. (1.1) in the energy space. However, in the radial case outside the origin, we reduce to the one-dimensional case with perturbation and could obtain the same results as for $N = 1$ (see [29] and also the extension by Hamza and Zaag in [16] to the Klein–Gordon equation and other damped lower-order perturbations of Eq. (1.4)). Recently, Merle and Zaag could address the higher-dimensional case in the subconformal case and prove the stability of the explicit selfsimilar profile with respect to the blow-up point and initial data (see [32,33]). Considering the behavior of radial solutions at the origin, Donninger and Schörkhuber were able to prove the stability of the ODE solution $u (t) = κ_{0} (p) {(T - t)}^{- \frac{2}{p - 1}}$ with respect to small perturbations in initial data, in the Sobolev subcritical range [9] and also in the supercritical range in [10]. Let us also mention that Killip, Stoval and Vişan proved in [20] that in superconformal and Sobolev subcritical range, an upper bound on the blow-up rate is available. This was further refined by Hamza and Zaag in [17].

The question of the perturbed non-linear wave equation was later investigated by Hamza and Zaag in [15] and [14] where they consider a class of perturbed equations, with $(H_{f})$ and $(H_{g})$ replaced by a more restrictive conditions: $| f (u) | ⩽ M (1 + | u |^{q})$ and $| g (u) | ⩽ M (1 + | u |)$ for some $q < p$ , $M > 0$ . Then, they proved a similar result to (1.5), valid in the subconformal case. Let us also mention that in [12], the authors extended the results obtained in [14] and [15] to strong perturbed equation of (1.1) satisfying $(H_{f})$ and $(H_{g})$ in the subconformal case ( $1 < p < p_{c}$ ). Recently, in [13] we extended the results known in [12] to the conformal case ( $p = p_{c}$ ) only for radial solutions, assuming that the parameter a satisfies $a > 2$ . However, these two assumptions appeared to us as technical and non-natural. As a matter of fact, coming with new ideas (the use of a Pohozaev identity), we aim in this work to remove the radial assumptions, though keeping the condition $a > 2$ . In fact, our main contribution in this paper is to construct a Lyapunov functional in similarity variables for the problem (1.1) in the non-radial case, relying on the use of a Pohozaev-type identity.

Pohozaev-type identity has been widely used in mathematics literature and the first results are due to [34], where among other things, he proved the non-existence of positive solutions for the elliptic equation $Δ u + | u |^{p - 1} u = 0$ , in the supercritical case. Later Giga and Kohn, in [11] characterize all stationary solutions in self-similar variables of non-linear heat equations $\partial_{t} u = Δ u + | u |^{p - 1} u$ , in the subcritical case. Recently the same type of identity have been used in the analysis of elliptic PDEs (see [8,36]). In our work, we construct a Pohozaev identity obtained by multiplying Eq. (1.7) by $y \cdot \nabla w$ in a suitable weighted space. As we see above, the use of this Pohozaev identity is crucial to construct a Lyapunov functional.

Let us introduce the following similarity variables, for any $(x_{0}, T_{0})$ such that $0 < T_{0} ⩽ T (x_{0})$ : $\begin{matrix} (1.6) & y = \frac{x - x_{0}}{T_{0} - t}, s = - log (T_{0} - t), u (x, t) = {(T_{0} - t)}^{- \frac{2}{p_{c} - 1}} w_{x_{0}, T_{0}} (y, s) . \end{matrix}$ From (1.1), the function $w_{x_{0}, T_{0}}$ (we write w for simplicity) satisfies the following equation for all $y \in B \equiv B (0, 1)$ and $s ⩾ - log T_{0}$ : $\begin{matrix} (1.7) & \begin{matrix} \partial_{s}^{2} w = & div (\nabla w - (y \cdot \nabla w) y) - \frac{2 (p_{c} + 1)}{{(p_{c} - 1)}^{2}} w + | w |^{p_{c} - 1} w - \frac{p_{c} + 3}{p_{c} - 1} \partial_{s} w \\ - 2 y \cdot \nabla \partial_{s} w + e^{\frac{- 2 p_{c} s}{p_{c} - 1}} f (e^{\frac{2 s}{p_{c} - 1}} w) + e^{- \frac{2 p_{c} s}{p_{c} - 1}} g (e^{\frac{(p_{c} + 1) s}{p_{c} - 1}} (\partial_{s} w + y \cdot \nabla w + \frac{2}{p_{c} - 1} w)) . \end{matrix} \end{matrix}$ This change of variables transforms the backward light cone with vortex $(x_{0}, T_{0})$ into the infinite cylinder $(y, s) \in B \times [- log T_{0}, + \infty)$ . In the new set of variables $(y, s)$ , the behavior of u as $t \to T_{0}$ is equivalent to the behavior of w as $s \to + \infty$ . In order to keep our analysis clear, we may assume that $f (u) = \frac{| u |^{p_{c}}}{{log}^{a} (2 + u^{2})}$ and $g \equiv 0$ , in Eq. (1.1) and refer the reader to [15] and [14] for straightforward adaptations to the general case where $f (u) ≢ \frac{| u |^{p_{c}}}{{log}^{a} (2 + u^{2})}$ and $g ≢ 0$ . Also, if $T_{0} = T (x_{0})$ , then we simply write $w_{x_{0}}$ instead of $w_{x_{0}, T (x_{0})}$ .

Equation (1.7) will be studied in the Hilbert space $H$ $\begin{matrix} H = {(w_{1}, w_{2}) | \int_{B} (w_{2}^{2} + | \nabla w_{1} |^{2} (1 - | y |^{2}) + w_{1}^{2}) d y < + \infty} . \end{matrix}$

In general, the treatment of the conformal case requires a new idea as compared to the subconformal case. In fact, the method of perturbation of the Lyapunov functional used in [15] and [12] works in the sub-conformal case but does not work in the conformal case. Let us recall that in [14], we studied the problem in the conformal case, if we replace $(H_{f})$ by a more restrictive condition: $\begin{matrix} (1.8) & | f (u) | ⩽ M (1 + | u |^{q}) for some q < p_{c} . \end{matrix}$ We proceeded in two steps to construct the Lyapunov functional: first, we exploited some functional to obtain a rough estimate to the blow-up solution namely an exponentially large bound. Even though this estimate seems bad, it was very useful to allow us to derive a natural Lyapunov functional for Eq. (1.7), a crucial step to derive the optimal estimate as in (1.5). Let us note that the method used in [14] under the restrictive condition (1.8) breaks down when our perturbation is stronger, namely when $f (u) \equiv \frac{| u |^{p_{c}}}{{log}^{a} (2 + u^{2})}$ . Let us mention that we overcome this difficulty with Saidi in [13] by proceeding in three steps to construct the Lyapunov functional: first, as in [14] we use some functional to obtain an exponentially large estimate for the blow-up solution. Then, we use this exponential bound to obtain a polynomial estimate. However, this step works only if the solution is radial. Finally this polynomial estimate allows us to prove that we have a natural Lyapunov functional for Eq. (1.7), valid only when $a > 2$ . Then, we derive the optimal result (1.5) if the solution is radial. That obstruction fully justifies our new paper, where we invent a new idea to get our optimal result for a non-radial blow-up solution of (1.7), when $a > 2$ .

Let us first recall the rough exponential space–time estimate of the solution u of (1.1) near any non-characteristic point obtained in [13]. More precisely, we established the following results:

Exponential space–time estimate of solution of (1.7).
Let u a solution of (1.1) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Then for all $η \in (0, 1)$ , there exists $t_{0} (x_{0}) \in [0, T (x_{0}))$ such that, for all $s ⩾ - log (T (x_{0}) - t_{0} (x_{0}))$ , we have $\begin{matrix} (1.9) & \int_{s}^{s + 1} \int_{B} ({| \nabla w_{x_{0}} (y, τ) |}^{2} + {| w_{x_{0}} (y, τ) |}^{p_{c} + 1}) d y d τ ⩽ K_{1} e^{η s}, \end{matrix}$ and $\begin{matrix} (1.10) & \int_{s}^{s + 1} \int_{B} \frac{{(\partial_{s} w_{x_{0}} (y, τ))}^{2}}{{(1 - | y |^{2})}^{1 - η}} d y d τ ⩽ K_{1} e^{\frac{p_{c} + 3}{2} η s}, \end{matrix}$ where the constant $K_{1}$ depends only on N, p, M, a, b, $δ_{0} (x_{0})$ , $T (x_{0})$ and $‖ (u (t_{0} (x_{0})), \partial_{t} u (t_{0} (x_{0}))) ‖_{H^{1} \times L^{2} (B (x_{0}, \frac{T (x_{0}) - t_{0} (x_{0})}{δ_{0} (x_{0})}))}$ .

In this paper, by exploiting a uniform version of the exponential estimates of (1.9) and (1.10) (see (2.1) and (2.2) below), we obtain the following polynomial space–time estimate:
Theorem 1.1 (A polynomially space–time estimate of solution of (1.7)).

Let u a solution of ( 1.1 ) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Then for all $b \in (1, a)$ there exist $t_{1} (x_{0}) = t_{1} (x_{0}, b) ⩾ t_{0} (x_{0})$ such that for all $s ⩾ - log (T (x_{0}) - t_{1} (x_{0}))$ , $\begin{matrix} (1.11) & \int_{s}^{s + 1} \int_{B} ({(\partial_{s} w_{x_{0}} (y, τ))}^{2} + {| \nabla w_{x_{0}} (y, τ) |}^{2} + {| w_{x_{0}} (y, τ) |}^{p_{c} + 1}) d y d τ ⩽ K_{2} s^{b}, \end{matrix}$ where the constant $K_{2}$ depends only on N, p, M, a, b, $K_{1}$ , $δ_{0} (x_{0})$ , $T (x_{0})$ and $‖ (u (t_{1} (x_{0})), \partial_{t} u (t_{1} (x_{0}))) ‖_{H^{1} \times L^{2} (B (x_{0}, \frac{T (x_{0}) - t_{1} (x_{0})}{δ_{0} (x_{0})}))}$ .

Theorem 1.1 can be written in the original variables in the following corollary:

Corollary 1.2.
Let u a solution of ( 1.1 ) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Then for all $b \in (1, a)$ there exist $t_{1} (x_{0}) = t_{1} (x_{0}, b) ⩾ t_{0} (x_{0})$ such that for all $t \in [t_{1} (x_{0}), T (x_{0}))$ , we have $\begin{matrix} \int_{T (x_{0}) - t}^{T (x_{0}) - \frac{t}{2}} \int_{B (x_{0}, T (x_{0}) - τ)} ({| \partial_{t} u (x, τ) |}^{2} + {| \nabla u (x, τ) |}^{2} + {| u (x, τ) |}^{p_{c} + 1}) d x d τ ⩽ K_{2} {| log (T (x_{0}) - t) |}^{b} . \end{matrix}$

Now, we are able to adapt the analysis performed in [10] for Eq. (1.7) and announce our main result valid only when $a > 2$ :
Theorem 1.3 (Blow-up rate for Eq. (1.1)).

Let $a > 2$ , consider u a solution of ( 1.1 ) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point, then there exist ${\hat{S}}_{2}$ large enough such that

For all $s ⩾ {\hat{s}}_{2} (x_{0}) = max ({\hat{S}}_{2}, - log \frac{T (x_{0})}{4})$ , $\begin{matrix} 0 < ε_{0} ⩽ {‖ w_{x_{0}} (s) ‖}_{H^{1} (B)} + {‖ \partial_{s} w_{x_{0}} (s) ‖}_{L^{2} (B)} ⩽ K, \end{matrix}$ where $w_{x_{0}, T (x_{0})}$ is defined in ( 1.6 ) and B is the unit ball of $R^{N}$ .

For all $t \in [t_{2} (x_{0}), T (x_{0}))$ , where $t_{2} (x_{0}) = T (x_{0}) - e^{- {\hat{s}}_{2} (x_{0})}$ , we have $\begin{array}{rcl} 0 & < & ε_{0} ⩽ {(T (x_{0}) - t)}^{\frac{2}{p_{c} - 1}} \frac{‖ u (t) ‖_{L^{2} (B (x_{0}, T (x_{0}) - t))}}{{(T (x_{0}) - t)}^{\frac{N}{2}}} \\ + {(T (x_{0}) - t)}^{\frac{2}{p_{c} - 1} + 1} (\frac{‖ \partial_{t} u (t) ‖_{L^{2} (B (x_{0}, T (x_{0}) - t))}}{{(T (x_{0}) - t)}^{\frac{N}{2}}} + \frac{‖ \nabla u (t) ‖_{L^{2} (B (x_{0}, T (x_{0}) - t))}}{{(T (x_{0}) - t)}^{\frac{N}{2}}}) ⩽ K, \end{array}$ where $K = K (K_{2}, N, p, M, a, b, T (x_{0}), t_{2} (x_{0}), ‖ (u (t_{2} (x_{0})), \partial_{t} u (t_{2} (x_{0}))) ‖_{H^{1} \times L^{2} (B (x_{0}, \frac{T (x_{0}) - t_{2} (x_{0})}{δ_{0} (x_{0})}))})$ .

Remark 1.1.
Please note that we crucially need a covering technique in our argument, that is why we need to prove a uniform version for x near $x_{0}$ (see the exponential space–time estimate written in (2.1) and (2.2), Theorem 1.1′). It happens that the generalization to a uniform version valid in the set ${(x, T_{0} - δ_{0} (x_{0}) (x - x_{0})), t_{2} (x_{0}) ⩽ T_{0} ⩽ T (x_{0}) and | x - x_{0} | ⩽ \frac{T_{0}}{δ_{0} (x_{0})}}$ is straightforward and we refer to [14] for more details.

This paper is organized as follows: In Section 2, we give a new decreasing functional for Eq. (1.7). Then we prove Theorem 1.1, where we obtain a polynomial space–time estimate of the solution w. Using this result, we prove in Section 3 that the “natural” functional is a Lyapunov functional for Eq. (1.7) with the additional assumption $a > 2$ . Then, proceeding as in [13], we prove Theorem 1.3.

We mention that C will be used to denote a constant that has depends on N, a, b and M which may vary from line to line. We also introduce $\begin{matrix} (1.12) & F (u) = \int_{0}^{u} f (v) d v . \end{matrix}$
2. Proof of Theorem 1.1

Note that our approach in this section is very close to [13]. In fact, we use an uniform version for x near $x_{0}$ for the exponential bound on time average to obtain an uniform version for x near $x_{0}$ polynomial estimate on time average of the $H^{1} \times L^{2} (B)$ norm of $(w, \partial_{s} w)$ . More precisely, this section is devoted to the proof of a general version of Theorem 1.1, uniform for x near $x_{0}$ (see Theorem 1.1′). This section is divided into four subsections:

In the first one we give some classical energy estimates following from the multiplication of Eq. (1.7) by $w {(1 - | y |^{2})}^{s^{- b}}$ and $\partial_{s} w {(1 - | y |^{2})}^{s^{- b}}$ .

The second subsection is devoted to give new energy estimates following from the multiplication of Eq. (1.7) by $y \cdot \nabla w {(1 - | y |^{2})}^{1 + s^{- b}} (1 - log (1 - | y |^{2}))$ .

By combining the above energy estimates obtained in the two subsections, we construct a decreasing functional for Eq. (1.7) and a blow-up criterion involving this functional.

Then, we conclude the proof of Theorem 1.1′.

Now, we start by stating the uniform version of the exponential bound on time average for x near

x_{0}

obtained in [13].

Uniform exponential space–time estimate of solution of (1.7).
Consider u a solution of (1.1) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ a non-characteristic point. Then for all $η \in (0, 1)$ , there exists $t_{0} (x_{0}) \in [0, T (x_{0}))$ such that, for all $T_{0} \in [t_{0} (x_{0}), T (x_{0})]$ , for all $s ⩾ - log (T_{0} - t_{0} (x_{0}))$ and $x \in R^{N}$ , where $| x - x_{0} | ⩽ \frac{e^{- s}}{δ_{0} (x_{0})}$ , we have $\begin{matrix} (2.1) & \int_{s}^{s + 1} \int_{B} ({| \nabla w (y, τ) |}^{2} + {| w (y, τ) |}^{p_{c} + 1}) d y d τ ⩽ K_{1} e^{η s}, \end{matrix}$ and $\begin{matrix} (2.2) & \int_{s}^{s + 1} \int_{B} \frac{{(\partial_{s} w (y, τ))}^{2}}{{(1 - | y |^{2})}^{1 - η}} d y d τ ⩽ K_{1} e^{\frac{p_{c} + 3}{2} η s}, \end{matrix}$ where $w = w_{x, T^{} (x)}$ is defined in (1.6) with $\begin{matrix} (2.3) & T^{} (x) = T_{0} - δ_{0} (x_{0}) (x - x_{0}), \end{matrix}$ $K_{1}$ depends on N, p, M, a, $δ_{0} (x_{0})$ , $T (x_{0})$ and $‖ (u (t_{0} (x_{0})), \partial_{t} u (t_{0} (x_{0}))) ‖_{H^{1} \times L^{2} (B (x_{0}, \frac{T (x_{0}) - t_{0} (x_{0})}{δ_{0} (x_{0})}))}$ .

Consider u a solution of (1.1) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Let $T_{0} \in (t_{0} (x_{0}), T (x_{0})]$ , for all $x \in R^{N}$ such that $| x - x_{0} | ⩽ \frac{T_{0}}{δ_{0} (x_{0})}$ , then we write w instead of $w_{x, T^{} (x)}$ defined in (1.6) with $T^{} (x)$ given in (2.3). As in [13], for any $b \in (1, a)$ , we put the equation in w in the following form: $\begin{array}{rcl} \partial_{s}^{2} w & = & \frac{1}{ϕ} div (ϕ \nabla w - (y \cdot \nabla w) ϕ y) + \frac{2}{s^{b}} y \cdot \nabla w - \frac{2 p_{c} + 2}{{(p_{c} - 1)}^{2}} w + | w |^{p_{c} - 1} w \\ (2.4) & - \frac{p_{c} + 3}{p_{c} - 1} \partial_{s} w - 2 y \cdot \nabla \partial_{s} w + e^{\frac{- 2 p_{c} s}{p_{c} - 1}} f (e^{\frac{2 s}{p_{c} - 1}} w) \forall y \in B and s ⩾ - log T^{*} (x), \end{array}$ where $\begin{matrix} (2.5) & ϕ = ϕ (y, s) = {(1 - | y |^{2})}^{s^{- b}} . \end{matrix}$

A key step is to find a functional $E (s)$ satisfying a differential inequality of type: $\begin{matrix} (2.6) & \frac{d}{d s} E (s) ⩽ - \frac{1}{s^{μ}} \int_{B} \frac{{(\partial_{s} w)}^{2}}{{(1 - | y |^{2})}^{1 - s^{- b}}} d y + \frac{C}{s^{μ}} E (s) for some μ > 1 . \end{matrix}$

In order to control the perturbative terms, we view Eq. (1.7) as a perturbation of the conformal case (corresponding to $ϕ \equiv 1$ already treated in [14]) with this term $\frac{2}{s^{b}} y \cdot \nabla w$ . Even the term is a lower order term with respect to the nonlinearity this term has a clear effect because an estimate of type (2.6) implies polynomial estimate on time average. It is worth noticing here that the weight $ϕ (y, s)$ defined in (2.5) (we write ϕ for simplicity), depends on time, it is not the case in this series of papers [14,15,17,24–29] and [12] we expect that the derivations in time are problematic. In fact, we note after observation, that there are new terms appearing compared with the previous works. Note that this problem was overcome in the radial case in [13], since the analysis uses the fact that the tangential part $\nabla_{θ} w = \nabla w - \frac{y \cdot \nabla w}{| y |^{2}} y$ of $\nabla w$ vanishes. Here, we further refine our argument allowing to handle the tangential part $\nabla_{θ} w$ to construct a function satisfying (2.6) in the non-radial case. Our method uses a new functional obtained by multiplying Eq. (2.4) by $y \cdot \nabla w {(1 - | y |^{2})}^{1 + s^{- b}} (1 - log (1 - | y |^{2}))$ . The addition of this functional to some energy estimates established by multiplying Eq. (1.7) by w and $\partial_{s} w$ in suitable weighted spaces permitted the control of the bad terms and is required even in the non-radial case.

Notice that in the rest of this section in spirit lightening the paper, we define $\begin{matrix} (2.7) & \nabla_{r} w = \frac{y \cdot \nabla w}{| y |^{2}} y and \nabla_{θ} w = \nabla w - \frac{y \cdot \nabla w}{| y |^{2}} y . \end{matrix}$ Then, it is given by (2.7), we can write $\nabla w = \nabla_{r} w + \nabla_{θ} w$ and we have the identities $\begin{matrix} (2.8) & | y |^{2} | \nabla w |^{2} - {(y \cdot \nabla w)}^{2} = | y |^{2} | \nabla_{θ} w |^{2}, \end{matrix}$ and $\begin{matrix} (2.9) & | \nabla w |^{2} - {(y \cdot \nabla w)}^{2} = | \nabla_{θ} w |^{2} + (1 - | y |^{2}) | \nabla_{r} w |^{2} . \end{matrix}$
2.1. Classical energy estimates

To control the norm of $(w (s), \partial_{s} w (s))$ , we start by introducing the following natural functionals, for all $b \in (1, a)$ , $\begin{array}{l} \begin{matrix} E (w (s), s) = & \int_{B} (\frac{1}{2} {(\partial_{s} w)}^{2} + \frac{1}{2} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) + \frac{p_{c} + 1}{{(p_{c} - 1)}^{2}} w^{2} - \frac{| w |^{p_{c} + 1}}{p_{c} + 1}) ϕ (y, s) d y \\ - e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} \int_{B} F (e^{\frac{2 s}{p_{c} - 1}} w) ϕ (y, s) d y, \end{matrix} \\ (2.10) & J (w (s), s) = - \frac{1}{s^{b}} \int_{B} w \partial_{s} w ϕ (y, s) d y + \frac{N}{2 s^{b}} \int_{B} w^{2} ϕ (y, s) d y, \\ H (w (s), s) = E (w (s), s) + J (w (s), s) . \end{array}$

In order to bound the time derivative of $H (w (s), s)$ , we begin with bounding the time derivative of $E (w (s), s)$ in the following lemma:

Lemma 2.1.
For all $b \in (1, a)$ , $ε \in (0, \frac{1}{2})$ and $s ⩾ max (- log T^{} (x), 1)$ , we have* $\begin{array}{rcl} \frac{d}{d s} (E (w (s), s)) & = & - \frac{2}{s^{b}} \int_{B} {(\partial_{s} w)}^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y + \frac{2}{s^{b}} \int_{B} y \cdot \nabla w \partial_{s} w ϕ (y, s) d y \\ + \frac{b}{s^{b + 1}} \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{- \frac{2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) ϕ (y, s) log (1 - | y |^{2}) d y \\ (2.11) & - \frac{b}{2 s^{b + 1}} \int_{B} | \nabla_{θ} w |^{2} | y |^{2} ϕ (y, s) log (1 - | y |^{2}) d y + Σ_{1} (s), \end{array}$ where $Σ_{1} (s)$ satisfies $\begin{array}{rcl} Σ_{1} (s) & ⩽ & \frac{b ε}{2 s^{b}} \int_{B} ({(\partial_{s} w)}^{2} + | \nabla w |^{2} (1 - | y |^{2}) + \frac{2 p_{c} + 2}{{(p_{c} - 1)}^{2}} w^{2}) ϕ (y, s) d y \\ (2.12) & + \frac{C}{s^{a}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y + C e^{- \frac{ε}{4} s} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y + C e^{- s} . \end{array}$
Proof.
Multiplying (2.4) by $\partial_{s} w ϕ (y, s)$ and integrating over the ball B, we obtain (2.11) where $Σ_{1} (s) = Σ_{1}^{1} (s) + Σ_{1}^{2} (s)$ and where $\begin{array}{l} (2.13) & Σ_{1}^{1} (s) = \frac{2 (p_{c} + 1)}{p_{c} - 1} e^{- \frac{2 (p_{c} + 1) s}{p_{c} - 1}} \int_{B} (F (e^{\frac{2 s}{p_{c} - 1}} w) - \frac{e^{\frac{2 s}{p_{c} - 1}}}{p_{c} + 1} w f (e^{\frac{2 s}{p_{c} - 1}} w)) ϕ (y, s) d y, \\ (2.14) & \begin{matrix} Σ_{1}^{2} (s) = & - \frac{b}{2 s^{b + 1}} \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) log (1 - | y |^{2}) d y \\ - \frac{b}{s^{b + 1}} \frac{p + 1}{{(p - 1)}^{2}} \int_{B} w^{2} ϕ (y, s) log (1 - | y |^{2}) d y \\ - \frac{b}{2 s^{b + 1}} \int_{B} | \nabla_{r} w |^{2} (1 - | y |^{2}) ϕ (y, s) log (1 - | y |^{2}) d y . \end{matrix} \end{array}$ Now, we control the terms $Σ_{1}^{1} (s)$ and $Σ_{1}^{2} (s)$ . Clearly the functions f and F defined in (1.12) satisfies the following estimate: $\begin{matrix} (2.15) & | F (x) | + | x f (x) | ⩽ C (1 + \frac{| x |^{p_{c} + 1}}{{log}^{a} (2 + x^{2})}) . \end{matrix}$ It easily follows from (2.13) and (2.15) that for all $s ⩾ max (- log T^{} (x), 1)$ , we write $\begin{matrix} (2.16) & Σ_{1}^{1} (s) ⩽ C \int_{B} \frac{| w |^{p_{c} + 1}}{{log}^{a} (2 + e^{\frac{4 s}{p_{c} - 1}} w^{2})} ϕ (y, s) d y + C e^{- s} . \end{matrix}$ As in [12] and [13], for all $s ⩾ max (- log T^{} (x), 1)$ , we divide the ball B into two parts $\begin{matrix} (2.17) & A_{1} (s) = {y \in B ∣ w^{2} (y, s) ⩽ e^{- \frac{2 s}{p_{c} - 1}}} and A_{2} (s) = {y \in B ∣ w^{2} (y, s) ⩾ e^{- \frac{2 s}{p_{c} - 1}}} . \end{matrix}$ Using the definition of the set $A_{1} (s)$ defined in (2.17) we get, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.18) & \int_{A_{1} (s)} \frac{| w |^{p_{c} + 1}}{{log}^{a} (2 + e^{\frac{4 s}{p_{c} - 1}} w^{2})} ϕ (y, s) d y ⩽ C e^{- \frac{(p_{c} + 1) s}{p_{c} - 1}} \int_{A_{1} (s)} ϕ (y, s) d y ⩽ C e^{- s} . \end{matrix}$ Also, by using the definition of the set $A_{2} (s)$ defined in (2.17), we can write if $y \in A_{2} (s)$ , we have $log (2 + e^{\frac{4 s}{p_{c} - 1}} w^{2}) ⩾ \frac{2 s}{p_{c} - 1}$ , one has, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.19) & \int_{A_{2} (s)} \frac{| w |^{p_{c} + 1}}{{log}^{a} (2 + e^{\frac{4 s}{p_{c} - 1}} w^{2})} ϕ (y, s) d y ⩽ \frac{C}{s^{a}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y . \end{matrix}$ Hence, the inequality (2.16), (2.17), (2.18) and (2.19), imply that $\begin{matrix} (2.20) & Σ_{1}^{1} (s) ⩽ \frac{C}{s^{a}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y + C e^{- s} for all s ⩾ max (- log T^{} (x), 1) . \end{matrix}$ Now, we are going to estimate $Σ_{1}^{2} (s)$ . The treatment of this term is more difficult because it contains terms with singular weight. In fact, unlike the terms in $Σ_{1}^{1} (s)$ which constituting by therms with the weight is $ϕ (y, s)$ , here the weight is $- ϕ (y, s) log (1 - | y |^{2})$ . To overcome this problem, we divide the unit ball B in two parts: a first part which is near the boundary $\partial B$ and the other is the rest of the unit ball. More precisely, let $ε \in (0, \frac{1}{2})$ , for all $s ⩾ max (- log T^{} (x), 1)$ , we divide B into two parts $\begin{matrix} (2.21) & B_{1} (s) = {y \in B ∣ 1 - | y |^{2} ⩽ e^{- ε s}} and B_{2} (s) = {y \in B ∣ 1 - | y |^{2} ⩾ e^{- ε s}} . \end{matrix}$ We see that: $Σ_{1}^{2} (s) = \underset{χ_{1} (s)}{\underset{︸}{χ_{1}^{1} (s) + χ_{1}^{2} (s) + χ_{1}^{3} (s)}} + χ_{2} (s)$ , where $\begin{array}{l} χ_{1}^{1} (s) = - \frac{b}{2 s^{b + 1}} \int_{B_{1} (s)} {(\partial_{s} w)}^{2} ϕ (y, s) log (1 - | y |^{2}) d y, \\ χ_{1}^{2} (s) = - \frac{b}{2 s^{b + 1}} \int_{B_{1} (s)} | \nabla_{r} w |^{2} (1 - | y |^{2}) ϕ (y, s) log (1 - | y |^{2}) d y, \\ χ_{1}^{3} (s) = - \frac{b}{s^{b + 1}} \frac{p_{c} + 1}{{(p_{c} - 1)}^{2}} \int_{B_{1} (s)} w^{2} ϕ (y, s) log (1 - | y |^{2}) d y, \end{array}$ and where $\begin{array}{rcl} χ_{2} (s) & = & - \frac{b}{2 s^{b + 1}} \int_{B_{2} (s)} ({(\partial_{s} w)}^{2} + \frac{2 p_{c} + 2}{{(p_{c} - 1)}^{2}} w^{2}) ϕ (y, s) log (1 - | y |^{2}) d y \\ - \frac{b}{2 s^{b + 1}} \int_{B_{2} (s)} | \nabla_{r} w |^{2} (1 - | y |^{2}) ϕ (y, s) log (1 - | y |^{2}) d y . \end{array}$ From the fact that, for all $s ⩾ max (- log T^{} (x), 1)$ , the function $y \mapsto ϕ (y, s) {(1 - | y |^{2})}^{\frac{1}{4}} log (1 - | y |^{2})$ is uniformly bounded on B and using the inequality $\begin{matrix} {(1 - | y |^{2})}^{\frac{1}{4}} ⩽ e^{- \frac{ε}{4} s} for all y \in B_{1} (s), \end{matrix}$ we can write, for all $s ⩾ max (- log T^{} (x), 1)$ , $\begin{matrix} (2.22) & χ_{1}^{1} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} \frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} d y . \end{matrix}$ Similarly we obtain easily, for all $s ⩾ max (- log T^{} (x), 1)$ , $\begin{matrix} (2.23) & χ_{1}^{2} (s) + χ_{1}^{3} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} | \nabla w |^{2} d y + C e^{- \frac{ε}{4} s} \int_{B} \frac{w^{2}}{\sqrt{1 - | y |^{2}}} d y . \end{matrix}$ Let us recall from [24] the following Hardy-type inequality $\begin{matrix} (2.24) & \int_{B} w^{2} \frac{| y |^{2}}{\sqrt{1 - | y |^{2}}} d y ⩽ C \int_{B} | \nabla w |^{2} {(1 - | y |^{2})}^{\frac{3}{2}} d y + C \int_{B} w^{2} \sqrt{1 - | y |^{2}} d y . \end{matrix}$ Using the fact that $\frac{w^{2}}{\sqrt{1 - | y |^{2}}} = \frac{| y |^{2} w^{2}}{\sqrt{1 - | y |^{2}}} + w^{2} \sqrt{1 - | y |^{2}}$ , we conclude that $\begin{matrix} (2.25) & χ_{1}^{2} (s) + χ_{1}^{3} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} w^{2} d y + C e^{- \frac{ε}{4} s} \int_{B} | \nabla w |^{2} d y . \end{matrix}$ Combining (2.22) and (2.25), one easily obtain $\begin{matrix} (2.26) & χ_{1} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y . \end{matrix}$ Next, by (2.9) and by exploiting the fact that if $y \in B_{2} (s)$ , we have $- log (1 - | y |^{2}) ⩽ ε s$ , we have for all $s ⩾ max (- log T^{} (x), 1)$ , $\begin{matrix} (2.27) & χ_{2} (s) ⩽ \frac{b ε}{2 s^{b}} \int_{B} ({(\partial_{s} w)}^{2} + | \nabla w |^{2} (1 - | y |^{2}) + \frac{2 p_{c} + 2}{{(p_{c} - 1)}^{2}} w^{2}) ϕ (y, s) d y . \end{matrix}$ Then we infer from (2.26), (2.27) and the identity $Σ_{1}^{2} (s) = χ_{1} (s) + χ_{2} (s)$ we have, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{array}{rcl} Σ_{1}^{2} (s) & ⩽ & \frac{b ε}{2 s^{b}} \int_{B} ({(\partial_{s} w)}^{2} + | \nabla w |^{2} (1 - | y |^{2}) + \frac{2 p_{c} + 2}{{(p_{c} - 1)}^{2}} w^{2}) ϕ (y, s) d y \\ (2.28) & + C e^{- \frac{ε}{4} s} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y . \end{array}$ The result (2.12) derives immediately from (2.20), (2.28) and the identity $Σ_{1} (s) = Σ_{1}^{1} (s) + Σ_{1}^{2} (s)$ , which ends the proof of Lemma 2.1. □

We are going to prove the following estimate to the functional $J (w (s), s)$ .
Lemma 2.2.
For all* $b \in (1, a)$ and $ε \in (0, \frac{1}{2})$ , there exists $S_{1} ⩾ 1$ such that for all $s ⩾ max (- log T^{} (x), S_{1})$ , we have the following inequality:* $\begin{array}{l} \frac{d}{d s} (J (w (s), s)) \\ = - \frac{p_{c} + 7}{4 s^{b}} \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) d y + \frac{p_{c} + 3}{2 s^{b}} H (w (s), s) \\ - \frac{p_{c} - 1}{4 s^{b}} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y - \frac{2}{s^{b}} \int_{B} \partial_{s} w y \cdot \nabla w ϕ (y, s) d y \\ (2.29) & - \frac{(p_{c} + 1)}{2 (p_{c} - 1) s^{b}} \int_{B} w^{2} ϕ (y, s) d y - \frac{p_{c} - 1}{2 (p_{c} + 1) s^{b}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y + Σ_{2} (s), \end{array}$ where $Σ_{2} (s)$ satisfies $\begin{array}{rcl} Σ_{2} (s) & ⩽ & \frac{C}{s^{b + 1}} \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) d y + \frac{34}{(p_{c} + 16) s^{b}} \int_{B} {(\partial_{s} w)}^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y \\ + \frac{p_{c} - 1}{8 s^{b}} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) d y + \frac{3 (p_{c} + 1)}{8 (p_{c} - 1) s^{b}} \int_{B} w^{2} ϕ (y, s) d y \\ (2.30) & + \frac{C}{s^{a + b}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y + C e^{- \frac{ε}{4} s} \int_{B} (w^{2} + | \nabla w |^{2}) d y + C e^{- s} . \end{array}$
Proof.
Note that $J (w (s), s)$ is a differentiable function and we get, for all $s ⩾ - log T^{} (x)$ $\begin{array}{rcl} \frac{d}{d s} (J (w (s), s)) & = & - \frac{1}{s^{b}} \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) d y - \frac{1}{s^{b}} \int_{B} w \partial_{s}^{2} w ϕ (y, s) d y \\ + (\frac{b}{s} + N) \frac{1}{s^{b}} \int_{B} w \partial_{s} w ϕ (y, s) d y + \frac{b}{s^{2 b + 1}} \int_{B} w \partial_{s} w log (1 - | y |^{2}) ϕ (y, s) d y \\ (2.31) & - \frac{N b}{2 s^{b + 1}} \int_{B} w^{2} ϕ (y, s) d y - \frac{N b}{2 s^{2 b + 1}} \int_{B} w^{2} ϕ (y, s) log (1 - | y |^{2}) d y . \end{array}$ According to Eq. (2.4), we obtain $\begin{array}{l} \frac{d}{d s} (J (w (s), s)) \\ = - \frac{2}{s^{b}} \int_{B} \partial_{s} w y \cdot \nabla w ϕ (y, s) d y - \frac{1}{s^{b}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y \\ - \frac{1}{s^{b}} \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) d y + \frac{1}{s^{b}} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y \\ (2.32) & + \frac{2 p_{c} + 2}{{(p_{c} - 1)}^{2}} \frac{1}{s^{b}} \int_{B} w^{2} ϕ (y, s) d y + Σ_{2}^{1} (s) + Σ_{2}^{2} (s) + Σ_{2}^{3} (s) + Σ_{2}^{4} (s) + Σ_{2}^{5} (s), \end{array}$ where $\begin{array}{l} Σ_{2}^{1} (s) = \frac{N (2 - b)}{2 s^{b + 1}} \int_{B} w^{2} ϕ (y, s) d y + \frac{b}{s^{b + 1}} \int_{B} w \partial_{s} w ϕ (y, s) d y, \\ Σ_{2}^{2} (s) = - \frac{2}{s^{3 b}} \int_{B} w^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y + \frac{4}{s^{2 b}} \int_{B} w \partial_{s} w \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y, \\ Σ_{2}^{3} (s) = - \frac{e^{\frac{- 2 p_{c} s}{p_{c} - 1}}}{s^{b}} \int_{B} w f (e^{\frac{2 s}{p_{c} - 1}} w) ϕ (y, s) d y, \\ Σ_{2}^{4} (s) = \frac{b}{s^{2 b + 1}} \int_{B} w \partial_{s} w ϕ (y, s) log (1 - | y |^{2}) d y, \\ Σ_{2}^{5} (s) = - \frac{N b}{2 s^{2 b + 1}} \int_{B} w^{2} ϕ (y, s) log (1 - | y |^{2}) d y . \end{array}$ According to the expression of $H (w (s), s)$ , with some straightforward computation we obtain (2.2) where $\begin{matrix} (2.33) & Σ_{2} (s) = Σ_{2}^{1} (s) + Σ_{2}^{2} (s) + Σ_{2}^{3} (s) + Σ_{2}^{4} (s) + Σ_{2}^{5} (s) + Σ_{2}^{6} (s) + Σ_{2}^{7} (s), \end{matrix}$ and where $\begin{array}{l} Σ_{2}^{6} (s) = \frac{p_{c} + 3}{2 s^{b}} e^{\frac{- 2 (p_{c} + 1) s}{p - 1}} \int_{B} F (e^{\frac{2 s}{p - 1}} w) ϕ (y, s) d y, \\ Σ_{2}^{7} (s) = \frac{p_{c} + 3}{2 s^{2 b}} \int_{B} w \partial_{s} w ϕ (y, s) d y - \frac{N (p_{c} + 1)}{4 s^{2 b}} \int_{B} w^{2} ϕ (y, s) d y . \end{array}$ We are going now to estimate the different terms of (2.33). Thanks to the classical inequality $a b ⩽ a^{2} + b^{2}$ , we conclude that for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.34) & Σ_{2}^{1} (s) + Σ_{2}^{7} (s) ⩽ \frac{C}{s^{b + 1}} \int_{B} ({(\partial_{s} w)}^{2} + w^{2}) ϕ (y, s) d y . \end{matrix}$ By the Cauchy–Schwarz inequality, we write for all $μ \in (0, 1)$ $\begin{matrix} (2.35) & Σ_{2}^{2} (s) ⩽ \frac{2 (1 - μ)}{s^{b}} \int_{B} {(\partial_{s} w)}^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y + \frac{2 μ}{(1 - μ) s^{3 b}} \int_{B} w^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y . \end{matrix}$ Let us recall from [24] the following Hardy-type inequality, for all $η \in (0, 1)$ $\begin{matrix} (2.36) & \int_{B} h^{2} \frac{| y |^{2} ρ_{η}}{1 - | y |^{2}} d y ⩽ \frac{1}{η^{2}} \int_{B} | \nabla h |^{2} (1 - | y |^{2}) ρ_{η} d y + \frac{N}{η} \int_{B} h^{2} ρ_{η} d y . \end{matrix}$ Then, from (2.36), it follows that $\begin{matrix} (2.37) & \int_{B} w^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y ⩽ s^{2 b} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) d y + N s^{b} \int_{B} w^{2} ϕ (y, s) d y . \end{matrix}$ From (2.35), (2.37) with $μ = \frac{p_{c} - 1}{p_{c} + 15}$ , we conclude that, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{array}{rcl} Σ_{2}^{2} (s) & ⩽ & \frac{32}{(p_{c} + 15) s^{b}} \int_{B} {(\partial_{s} w)}^{2} \frac{| y |^{2} ϕ (y, s)}{1 - | y |^{2}} d y + \frac{p_{c} - 1}{8 s^{b}} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) d y \\ (2.38) & + \frac{C}{s^{2 b}} \int_{B} w^{2} ϕ (y, s) d y . \end{array}$ The same type of estimates used to obtain (2.20) are used here to deduce easily, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.39) & Σ_{2}^{3} (s) + Σ_{2}^{6} (s) ⩽ \frac{C}{s^{a + b}} \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y + C e^{- s} . \end{matrix}$ Furthermore by using the inequality $a b ⩽ a^{2} + b^{2}$ we write, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} Σ_{2}^{4} (s) ⩽ \frac{(p_{c} + 1)}{4 (p_{c} - 1) s^{b}} \int_{B} w^{2} ϕ (y, s) d y + \frac{C}{s^{3 b + 2}} \int_{B} {(\partial_{s} w)}^{2} {(log (1 - | y |^{2}))}^{2} ϕ (y, s) d y . \end{matrix}$ Using the fact that, the function $y \mapsto | y |^{- 2} \sqrt{1 - | y |^{2}} {(log (1 - | y |^{2}))}^{2}$ is bounded on B, we obtain $\begin{matrix} (2.40) & Σ_{2}^{4} (s) ⩽ \frac{(p_{c} + 1)}{4 (p_{c} - 1) s^{b}} \int_{B} w^{2} ϕ (y, s) d y + \frac{C}{s^{3 b + 2}} \int_{B} {(\partial_{s} w)}^{2} \frac{| y |^{2} ϕ (y, s)}{\sqrt{1 - | y |^{2}}} d y . \end{matrix}$ Finally, for all $s ⩾ max (- log T^{} (x), 1)$ , we are going to estimate $Σ_{2}^{5} (s)$ . For this, we divide B into two parts $B_{1} (s)$ and $B_{2} (s)$ as defined in (2.21). We write $Σ_{2}^{5} (s) = χ_{3} (s) + χ_{4} (s)$ , where $\begin{array}{l} χ_{3} (s) = - \frac{N b}{2 s^{2 b + 1}} \int_{B_{1} (s)} w^{2} ϕ (y, s) log (1 - | y |^{2}) d y, \\ χ_{4} (s) = - \frac{N b}{2 s^{2 b + 1}} \int_{B_{2} (s)} w^{2} ϕ (y, s) log (1 - | y |^{2}) d y . \end{array}$ Hence, if $y \in B_{1} (s)$ , then ${(1 - | y |^{2})}^{\frac{1}{4}} ⩽ e^{- \frac{ε}{4} s}$ , taking into account the fact that, for all $s ⩾ max (s_{0}, 1)$ , the function $y \mapsto | y |^{- 2} ϕ (y, s) {(1 - | y |^{2})}^{\frac{1}{4}} log (1 - | y |^{2})$ is uniformly bounded on B, we can write for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.41) & χ_{3} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} w^{2} \frac{| y |^{2}}{\sqrt{1 - | y |^{2}}} d y . \end{matrix}$ Furthermore, thanks to the Hardy–Sobolev inequality (2.24), we conclude that $\begin{matrix} (2.42) & χ_{3} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} (w^{2} + | \nabla w |^{2}) d y . \end{matrix}$ Taking in consideration the fact that, if $y \in B_{2} (s)$ , we have $- log (1 - | y |^{2}) ⩽ ε s$ , we obtain $\begin{matrix} (2.43) & χ_{4} (s) ⩽ \frac{C}{s^{2 b}} \int_{B} w^{2} ϕ (y, s) d y . \end{matrix}$ By adding (2.42) and (2.43), we write $\begin{matrix} (2.44) & Σ_{2}^{5} (s) ⩽ \frac{C}{s^{2 b}} \int_{B} w^{2} ϕ (y, s) d y + C e^{- \frac{ε}{4} s} \int_{B} (w^{2} + | \nabla w |^{2}) d y . \end{matrix}$ Consequently, collecting (2.34), (2.38), (2.39), (2.40) and (2.44), one easily there exists $S_{1} ⩾ 1$ such that we obtain $Σ_{2} (s)$ satisfies (2.30), which end the proof of Lemma 2.2. □

Lemmas 2.1 and 2.2 allows to prove the following lemma: Lemma 2.3.
For all* $b \in (1, a)$ and $0 < ε ⩽ \frac{p_{c} - 1}{32 b (p_{c} + 1)}$ , there exist $S_{2} ⩾ S_{1}$ and $λ_{0} > 0$ such that for all $s ⩾ max (- log T^{} (x), S_{2})$ , we have the following inequality:* $\begin{array}{rcl} \frac{d}{d s} (H (w (s), s)) & ⩽ & \frac{p_{c} + 3}{2 s^{b}} H (w (s), s) - \frac{b}{2 s^{b + 1}} \int_{B} | \nabla_{θ} w |^{2} | y |^{2} ϕ (y, s) log (1 - | y |^{2}) d y \\ + \frac{b}{s^{b + 1}} \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{- \frac{2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) ϕ (y, s) log (1 - | y |^{2}) d y \\ - \frac{λ_{0}}{s^{b}} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y \\ (2.45) & - \frac{λ_{0}}{s^{b}} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{1 - | y |^{2}} + | w |^{p_{c} + 1} + w^{2}) ϕ (y, s) d y + Σ_{3} (s), \end{array}$ where $Σ_{3} (s)$ satisfies $\begin{matrix} (2.46) & Σ_{3} (s) ⩽ C e^{- \frac{ε}{4} s} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y + C e^{- s} . \end{matrix}$
Remark 2.1.
We notice that the term $- \frac{b}{2 s^{b + 1}} \int_{B} | \nabla_{θ} w |^{2} | y |^{2} ϕ (y, s) log (1 - | y |^{2}) d y$ in the inequality (2.45) is non-negative, which does not allow to construct a decreasing functional for Eq. (2.4). Please note that in [13] we only treat the radial solutions where this term vanishes. One main reason for this restriction is that we did not know control this term in the case of non-radial solutions. Here, let us recall that we consider the non-radial case that is why we need a new idea to overcome this problem. In fact, we construct a new functional $L (w (s), s)$ which is crucial to obtain a decreasing functional for Eq. (2.4) later.

2.2. New energy estimates

In this subsection, we start by introduce the crucial new functional $L (w (s), s)$ defined by the following: $\begin{matrix} (2.47) & L (w (s), s) = \int_{B} ({(y \cdot \nabla w)}^{2} + y \cdot \nabla w \partial_{s} w) ψ (y, s) d y, \end{matrix}$ where $ψ (y, s) = (1 - | y |^{2}) ϕ (y, s) (1 - log (1 - | y |^{2}))$ . As one can see in the statement below, this quantity arises from a Pohozaev identity obtained through the multiplication of Eq. (1.7) by $y \cdot \nabla w$ . This is the main novelty of our paper. More precisely, to estimate the time derivative of the functional $L (w (s), s)$ , we claim the following:

Lemma 2.4.
For all $b \in (1, a)$ and $s ⩾ max (- log T^{} (x), 1)$ , we have* $\begin{array}{rcl} \frac{d}{d s} L (w (s), s) & = & (1 + \frac{1}{s^{b}}) \int_{B} | \nabla_{θ} w |^{2} | y |^{2} ϕ (y, s) log (1 - | y |^{2}) d y \\ - (2 + \frac{2}{s^{b}}) \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) ϕ (y, s) log (1 - | y |^{2}) d y \\ (2.48) & + Σ_{3} (s), \end{array}$ where $\begin{array}{rcl} Σ_{3} (s) & ⩽ & C \int_{B} (\frac{{(\partial_{s} w)}^{2}}{1 - | y |^{2}} + | \nabla w |^{2} - {(y \cdot \nabla w)}^{2} + w^{2} + | w |^{p_{c} + 1}) ϕ (y, s) d y \\ (2.49) & + C e^{- \frac{s}{4}} \int_{B} | \nabla w |^{2} d y + C e^{- s} . \end{array}$
Proof.
Note that $L (w (s), s)$ is a differentiable function and we get, for all $s ⩾ - log T^{} (x)$ $\begin{matrix} (2.50) & \frac{d}{d s} L (w (s), s) = \int_{B} y \cdot \nabla w (\partial_{s}^{2} w + 2 y \cdot \nabla \partial_{s} w) ψ (y, s) d y + Σ_{3}^{1} (s) + Σ_{3}^{2} (s), \end{matrix}$ where $\begin{array}{l} Σ_{3}^{1} (s) = \int_{B} y \cdot \nabla \partial_{s} w \partial_{s} w ψ (y, s) d y, \\ Σ_{3}^{2} (s) = - \frac{b}{s^{b + 1}} \int_{B} ({(y \cdot \nabla w)}^{2} + y \cdot \nabla w \partial_{s} w) ψ (y, s) log (1 - | y |^{2}) d y . \end{array}$ By using (1.7) and integrating by parts, we have $\begin{array}{rcl} \frac{d}{d s} L (w (s), s) & = & \int_{B} y \cdot \nabla w div (\nabla w - (y \cdot \nabla w) y) ψ (y, s) d y \\ - \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) div (ψ (y, s) y) d y \\ (2.51) & + Σ_{3}^{1} (s) + Σ_{3}^{2} (s) + Σ_{3}^{3} (s), \end{array}$ where $\begin{matrix} (2.52) & Σ_{3}^{3} (s) = \frac{p_{c} + 1}{{(p_{c} - 1)}^{2}} \int_{B} w^{2} div (ψ (y, s) y) d y - \frac{p_{c} + 3}{p_{c} - 1} \int_{B} y \cdot \nabla w \partial_{s} w ψ (y, s) d y . \end{matrix}$ A straightforward computation yields the identity $\begin{matrix} (2.53) & div (ψ (y, s) y) = (N + 2 + \frac{2}{s^{b}}) ψ (y, s) - (2 + \frac{2}{s^{b}}) \frac{ψ (y, s)}{1 - | y |^{2}} + 2 | y |^{2} ϕ (y, s) . \end{matrix}$ From (2.53), we obtain $\begin{array}{l} - \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) div (ψ (y, s) y) d y \\ (2.54) & = - (2 + \frac{2}{s^{b}}) \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) ϕ (y, s) log (1 - | y |^{2}) d y + Σ_{3}^{4} (s), \end{array}$ where $\begin{array}{rcl} Σ_{3}^{4} (s) & = & - (N + 2 + \frac{2}{s^{b}}) \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) ψ (y, s) d y \\ (2.55) & + 2 \int_{B} (\frac{| w |^{p_{c} + 1}}{p_{c} + 1} + e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} F (e^{\frac{2 s}{p_{c} - 1}} w)) (1 + \frac{1}{s^{b}} - | y |^{2}) ϕ (y, s) d y . \end{array}$ After some simple integration by parts that we leave to the Appendix, (2.51) and (2.2), we obtain (2.48) where $\begin{matrix} (2.56) & Σ_{3} (s) = Σ_{3}^{1} (s) + Σ_{3}^{2} (s) + Σ_{3}^{3} (s) + Σ_{3}^{4} (s) + Σ_{3}^{5} (s), \end{matrix}$ and where $\begin{array}{rcl} Σ_{3}^{5} (s) & = & - \frac{1}{s^{b}} \int_{B} | \nabla_{θ} w |^{2} | y |^{2} ϕ (y, s) d y + \frac{N - 2}{2} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ψ (y, s) d y \\ (2.57) & + \frac{1}{s^{b}} \int_{B} {(y \cdot \nabla w)}^{2} ψ (y, s) d y - \int_{B} (1 - | y |^{2}) {(y \cdot \nabla w)}^{2} ϕ (y, s) d y . \end{array}$ Now, we control all the terms on the right-hand side of the identity (2.56). After integration by parts, we use (2.53) to show $\begin{matrix} Σ_{3}^{1} (s) = - \frac{N}{2} \int_{B} {(\partial_{s} w)}^{2} ψ (y, s) d y + (1 + \frac{1}{s^{b}}) \int_{B} {(\partial_{s} w)}^{2} \frac{| y |^{2} ψ (y, s)}{1 - | y |^{2}} d y - \int_{B} | y |^{2} {(\partial_{s} w)}^{2} ϕ (y, s) d y . \end{matrix}$ To estimate $Σ_{3}^{1} (s)$ , we using the fact that $0 ⩽ ψ (y, s) ⩽ C ϕ (y, s)$ , for all $y \in B$ , to write for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.58) & Σ_{3}^{1} (s) ⩽ C \int_{B} \frac{{(\partial_{s} w)}^{2}}{1 - | y |^{2}} ϕ (y, s) d y . \end{matrix}$ Note that by using the inequality $0 ⩽ - ψ (y, s) log (1 - | y |^{2}) ⩽ C ϕ (y, s)$ , for all $y \in B$ and the fact that $a b ⩽ a^{2} + b^{2}$ , we write for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.59) & Σ_{3}^{2} (s) ⩽ \underset{χ_{5} (s)}{\underset{︸}{- \frac{C}{s^{b + 1}} \int_{B} | \nabla w |^{2} ψ (y, s) log (1 - | y |^{2}) d y}} + \frac{C}{s^{b + 1}} \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) d y . \end{matrix}$ We would like now to find an estimate from the term $χ_{5} (s)$ . For this, we divide B into two parts $B_{3} (s)$ and $B_{4} (s)$ defined for all $s ⩾ max (- log T^{} (x), 1)$ by $\begin{matrix} (2.60) & B_{3} (s) = {y \in B ∣ 1 - | y |^{2} ⩽ e^{- s}} and B_{4} (s) = {y \in B ∣ 1 - | y |^{2} ⩾ e^{- s}} . \end{matrix}$ We write $χ_{5} (s) = χ_{5}^{1} (s) + χ_{5}^{2} (s)$ , where $\begin{array}{l} χ_{5}^{1} (s) = - \frac{C}{s^{b + 1}} \int_{B_{3} (s)} | \nabla w |^{2} ψ (y, s) log (1 - | y |^{2}) d y, \\ χ_{5}^{2} (s) = - \frac{C}{s^{b + 1}} \int_{B_{4} (s)} | \nabla w |^{2} ψ (y, s) log (1 - | y |^{2}) d y . \end{array}$ From the fact that, for all $s ⩾ max (- log T^{} (x), 1)$ the function $y \mapsto ϕ (y, s) \sqrt{1 - | y |^{2}} (1 - log (1 - | y |^{2})) log (1 - | y |^{2})$ is uniformly bounded on B and using the inequality $\begin{matrix} \sqrt{1 - | y |^{2}} ⩽ e^{- \frac{s}{2}} \forall y \in B_{3} (s), \end{matrix}$ we can write, for all $s ⩾ max (- log T^{} (x), 1)$ , $\begin{matrix} (2.61) & χ_{5}^{1} (s) ⩽ C e^{- \frac{s}{4}} \int_{B} | \nabla w |^{2} d y . \end{matrix}$ Next, if $y \in B_{4} (s)$ , we can write $- (1 - log (1 - | y |^{2})) log (1 - | y |^{2}) ⩽ C s^{2}$ , then we have for all $s ⩾ max (- log T^{} (x), 1)$ , $\begin{matrix} (2.62) & χ_{5}^{2} (s) ⩽ \frac{C}{s^{b - 1}} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) d y . \end{matrix}$ By using (2.61) and (2.62), we can write for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.63) & χ_{5} (s) ⩽ C e^{- \frac{s}{4}} \int_{B} | \nabla w |^{2} d y + \frac{C}{s^{b - 1}} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) d y . \end{matrix}$ By combining (2.59) and (2.63) we conclude, for all $s ⩾ max (- log T^{} (x), 1)$ , $\begin{matrix} (2.64) & Σ_{3}^{2} (s) ⩽ \frac{C}{s^{b - 1}} \int_{B} (| \nabla w |^{2} (1 - | y |^{2}) + {(\partial_{s} w)}^{2}) ϕ (y, s) d y + C e^{- \frac{s}{4}} \int_{B} | \nabla w |^{2} d y . \end{matrix}$ Adding (2.52) to the identity (2.53), we can write $\begin{array}{rcl} Σ_{3}^{3} (s) & = & N \frac{(p_{c} + 1)}{{(p_{c} - 1)}^{2}} \int_{B} w^{2} ψ (y, s) d y - 2 (1 + \frac{1}{s^{b}}) \frac{(p_{c} + 1)}{{(p_{c} - 1)}^{2}} \int_{B} w^{2} \frac{| y |^{2} ψ (y, s)}{1 - | y |^{2}} d y \\ (2.65) & + 2 \frac{(p_{c} + 1)}{{(p_{c} - 1)}^{2}} \int_{B} | y |^{2} w^{2} ϕ (y, s) d y \underset{χ_{6} (s)}{\underset{︸}{- \frac{p_{c} + 3}{p_{c} - 1} \int_{B} y \cdot \nabla w \partial_{s} w ψ (y, s) d y}} . \end{array}$ Note that, by using inequality $a b ⩽ a^{2} + b^{2}$ we obtain $\begin{matrix} (2.66) & χ_{6} (s) ⩽ \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ψ (y, s) (1 - log (1 - | y |^{2})) d y + C \int_{B} {(\partial_{s} w)}^{2} ϕ (y, s) d y . \end{matrix}$ From (2.65), (2.66), the fact that, for all $y \in B$ , we have $ψ (y, s) ⩽ C ϕ (y, s)$ and for all $s ⩾ max (- log T^{} (x), 1)$ the function $y \mapsto (1 - | y |^{2}) {(1 - log (1 - | y |^{2}))}^{2}$ is uniformly bounded on B, we write $\begin{matrix} (2.67) & Σ_{3}^{3} (s) ⩽ C \int_{B} ({(\partial_{s} w)}^{2} + w^{2} + | \nabla w |^{2} (1 - | y |^{2})) ϕ (y, s) d y . \end{matrix}$ The same type of estimates used to obtain (2.20) are used here together with the inequality $ψ (y, s) ⩽ C ϕ (y, s)$ to deduce easily, for all $s ⩾ max (- log T^{} (x), 1)$ $\begin{matrix} (2.68) & Σ_{3}^{4} (s) ⩽ C \int_{B} | w |^{p_{c} + 1} ϕ (y, s) d y + C e^{- s} . \end{matrix}$ Finally, it remains only to control the term $Σ_{3}^{5} (s)$ . Also, by using the fact that $ψ (y, s) ⩽ C ϕ (y, s)$ , we have $\begin{array}{rcl} Σ_{3}^{5} (s) & ⩽ & C \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y \\ (2.69) & \underset{χ_{7} (s)}{\underset{︸}{- \frac{1}{s^{b}} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) log (1 - | y |^{2}) d y}} \forall s ⩾ max (- log T^{} (x), 1) . \end{array}$ In a similar way to the treatment of $χ_{1}^{2} (s)$ and $χ_{2} (s)$ to (2.23) and (2.27), we write $\begin{matrix} (2.70) & χ_{7} (s) ⩽ C e^{- \frac{s}{4}} \int_{B} | \nabla w |^{2} d y + \frac{C}{s^{b - 1}} \int_{B} | \nabla w |^{2} (1 - | y |^{2}) ϕ (y, s) . \end{matrix}$ By adding (2.69) and (2.70), we get for all $s ⩾ max (- log T^{*} (x), 1)$ $\begin{matrix} (2.71) & Σ_{3}^{5} (s) ⩽ C \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y + C e^{- \frac{s}{4}} \int_{B} | \nabla w |^{2} d y . \end{matrix}$ Now, we are able to conclude the proof of the inequality (2.49). For this, we combine (2.56), (2.58), (2.64), (2.67), (2.68) and (2.71) to get the desired estimate (2.49) which ends the proof of Lemma 2.4. □

2.3. Existence of a decreasing functional for Eq. (2.4)

In this subsection, by using Lemmas 2.3 and 2.4, we are going to construct a decreasing functional for Eq. (2.4). Let us define the following functional: $\begin{matrix} (2.72) & N (w (s), s) = exp (\frac{p_{c} + 3}{(b - 1) s^{b - 1}}) K (w (s), s) + σ e^{- \frac{ε_{0}}{8} s}, \end{matrix}$ where σ is a constant will be determined later, $ε_{0} = \frac{p_{c} - 1}{32 b (p_{c} + 1)}$ and where $\begin{matrix} (2.73) & K (w (s), s) = H (w (s), s) + \frac{b}{2 (s + s^{b + 1})} L (w (s), s) . \end{matrix}$ We now state the following proposition:

Proposition 2.5.
For all $b \in (1, a)$ and $0 < ε < \frac{p_{c} - 1}{32 b (p_{c} + 1)}$ , there exists $S_{3} ⩾ S_{2}$ and $λ_{1} > 0$ , such that for all $s ⩾ max (- log T^{} (x), S_{3})$ , we have the following inequality:* $\begin{array}{rcl} N (w (s + 1), s + 1) - N (w (s), s) & ⩽ & - \frac{λ_{1}}{s^{b}} \int_{s}^{s + 1} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{1 - | y |^{2}} + w^{2} + | w |^{p_{c} + 1}) ϕ (y, τ) d y d τ \\ (2.74) & - \frac{λ_{1}}{s^{b}} \int_{s}^{s + 1} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, τ) d y d τ . \end{array}$ Moreover, there exists $S_{4} ⩾ S_{3}$ such that for all $s ⩾ max (- log T^{} (x), S_{4})$ , we have* $\begin{matrix} (2.75) & N (w (s), s) ⩾ 0 . \end{matrix}$
Proof.
Let $b \in (1, a)$ and $0 < ε < \frac{p_{c} - 1}{32 b (p_{c} + 1)}$ . Combining the Lemmas 2.3 and 2.4, the fact that $0 ⩽ ψ (y, s) ⩽ ϕ (y, s)$ and choose $S_{3} ⩾ S_{2}$ large enough there exist $μ_{1} > 0$ such that for all $s ⩾ max (- log T^{} (x), S_{3})$ , we have $\begin{array}{rcl} \frac{d}{d s} K (w (s), s) & ⩽ & \frac{p_{c} + 3}{2 s^{b}} K (w (s), s) - \frac{μ_{1}}{s^{b}} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y \\ - \frac{μ_{1}}{s^{b}} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{1 - | y |^{2}} + | w |^{p_{c} + 1} + w^{2}) ϕ (y, s) d y \\ (2.76) & + C e^{- \frac{ε}{4} s} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y + C e^{- s} . \end{array}$ Since, for all $s ⩾ 1$ , we have $C^{- 1} ⩽ exp (\frac{p_{c} + 3}{(b - 1) s^{b - 1}}) ⩽ C$ . Then there exist $μ_{2} > 0$ such that for all $s ⩾ max (- log T^{} (x), S_{3})$ , we have $\begin{array}{rcl} \frac{d}{d s} N (w (s), s) & ⩽ & - \frac{μ_{2}}{s^{b}} \int_{B} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) ϕ (y, s) d y \\ - \frac{μ_{2}}{s^{b}} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{1 - | y |^{2}} + | w |^{p_{c} + 1} + w^{2}) ϕ (y, s) d y \\ (2.77) & + C e^{- \frac{ε}{4} s} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y + C e^{- s} - \frac{ε_{0} σ}{8} e^{- \frac{ε_{0}}{8} s} . \end{array}$ We now exploit the exponential space–time estimates (2.1) and (2.2) in the particular case where $η = \frac{ε_{0}}{4 (p_{c} + 3)}$ , to show that $\begin{matrix} (2.78) & C e^{- \frac{ε_{0}}{4} s} \int_{s}^{s + 1} \int_{B} (\frac{{(\partial_{s} w)}^{2}}{\sqrt{1 - | y |^{2}}} + | \nabla w |^{2} + w^{2}) d y d τ ⩽ C e^{- \frac{ε_{0}}{8} s} . \end{matrix}$ By integrating the inequality (2.77) in time between s and $s + 1$ and taking into account $ε \in (0, ε_{0})$ , (2.78) and choose σ large enough to deduce (2.74).

To end the proof of the last point of Proposition 2.5, we refer the reader to [25] and [12]. Let us mention that our proof strongly relies on the fact that $p < 1 + \frac{4}{N - 2}$ which is implied by the fact that $p = p_{c} \equiv 1 + \frac{4}{N - 1}$ . □

2.4. Proof of Theorem 1.1′

We define the following time: $\begin{matrix} (2.79) & t_{1} (x_{0}) = max (T (x_{0}) - e^{- S_{4}}, 0) . \end{matrix}$ According to the Proposition 2.5, we obtain the following corollary which summarizes the principle properties of $N (w (s), s)$ defined in (2.72).

Corollary 2.6 (Estimate on $N (w (s), s)$ ).

For all $b \in (1, a)$ , there exists $t_{1} (x_{0}) \in [t_{0} (x_{0}), T (x_{0}))$ such that, for all $T_{0} \in (t_{1} (x_{0}), T (x_{0})]$ for all $s ⩾ - log (T_{0} - t_{1} (x_{0}))$ and $x \in R^{N}$ where $| x - x_{0} | ⩽ \frac{e^{- s}}{δ_{0} (x_{0})}$ , we have $\begin{array}{l} - C ⩽ N (w (s), s) ⩽ N (w ({\tilde{s}}_{0}), {\tilde{s}}_{0}), \\ \int_{s}^{s + 1} \int_{B} \frac{{(\partial_{s} w (y, τ))}^{2}}{1 - | y |^{2}} ϕ (y, s) d y d τ ⩽ C (1 + N (w ({\tilde{s}}_{0}), {\tilde{s}}_{0})) s^{b}, \\ \int_{s}^{s + 1} \int_{B_{1 / 2}} ({| \nabla w (y, τ) |}^{2} + {| w (y, τ) |}^{p_{c} + 1}) d y d τ ⩽ C (1 + N (w ({\tilde{s}}_{0}), {\tilde{s}}_{0})) s^{b}, \end{array}$ where ${\tilde{s}}_{0} = - log (T^{*} (x) - t_{1} (x_{0}))$ .

Remark 2.2.
Using the definition of (1.6) of $w_{x, T^{*} (x)} = w$ , we write easily $\begin{matrix} N (w ({\tilde{s}}_{0}), {\tilde{s}}_{0}) ⩽ {\tilde{K}}_{0}, \end{matrix}$ where ${\tilde{K}}_{0} = {\tilde{K}}_{0} (T (x_{0}) - t_{1} (x_{0}), ‖ (u (t_{1} (x_{0})), \partial_{t} u (t_{1} (x_{0}))) ‖_{H^{1} \times L^{2} (B (x_{0}, \frac{T (x_{0}) - t_{1} (x_{0})}{δ_{0} (x_{0})}))})$ .

With Corollary 2.6, we are in a position to state and prove Theorem 1.1′, which is a uniform version of Theorem 1.1 for x near $x_{0}$ . Theorem 1.1′ (Uniform polynomially space–time estimate of solution of (1.7)).

Let u a solution of (1.1) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Then for all $b \in (1, a)$ , for all $T_{0} \in [t_{1} (x_{0}), T (x_{0})]$ , for all $s ⩾ - log (T_{0} - t_{1} (x_{0}))$ and $x \in R^{N}$ , where $| x - x_{0} | ⩽ \frac{e^{- s}}{δ_{0} (x_{0})}$ , we have $\begin{matrix} (2.80) & \int_{s}^{s + 1} \int_{B} ({(\partial_{s} w (y, τ))}^{2} + {| \nabla w (y, τ) |}^{2} + {| w (y, τ) |}^{p_{c} + 1}) d y d τ ⩽ K_{2} s^{b}, \end{matrix}$ where $w = w_{x, T^{*} (x)}$ , where $T^{*} (x)$ is defined (2.3) and where the constant $K_{2}$ depends only on N, p, M, $K_{1}$ , $δ_{0} (x_{0})$ , $T (x_{0})$ and $‖ (u (t_{1} (x_{0})), \partial_{t} u (t_{1} (x_{0}))) ‖_{H^{1} \times L^{2} (B (x_{0}, \frac{T (x_{0}) - t_{1} (x_{0})}{δ_{0} (x_{0})}))}$ .

Proof.
Note that the estimate on the space–time $L^{2}$ norm of $\partial_{s} w$ was already proved in Corollary 2.6. Thus we focus on the space–time $L^{p_{c} + 1}$ norm of w and $L^{2}$ norm of $\nabla w$ . This estimate was proved in Corollary 2.6 but just for the space–time $L^{p_{c} + 1}$ norm of w and $L^{2}$ norm of $\nabla w$ in $B_{1 / 2}$ . To extend this estimate from $B_{1 / 2}$ to B we refer the reader to Merle and Zaag [25] (unperturbed case) and Hamza and Zaag [14] (perturbed case), where they introduce a new covering argument to extend the estimate of any known space $L^{q}$ norm of w, $\partial_{s} w$ or $\nabla w$ , from $B_{1 / 2}$ to B. □

3. Proof of Theorem 1.3

This section is devoted to the proof of Theorem 1.3. Throughout this section, we assume $a > 2$ . This section is divided into two parts:

In Section 3.1, based upon Theorem 1.1′, we construct a Lyapunov functional for Eq. (1.7) and a blow-up criterion involving this functional.

In Section 3.2, we prove Theorem 1.3.

3.1. Existence of a Lyapunov functional for Eq. (1.7) and a blow-up criterion

Consider u a solution of (1.1) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Let $T_{0} \in (t_{1} (x_{0}), T (x_{0})]$ , for all $x \in R^{N}$ such that $| x - x_{0} | ⩽ \frac{T_{0}}{δ_{0} (x_{0})}$ , then we write w instead of $w_{x, T^{*} (x)}$ defined in (1.6) with $T^{*} (x)$ given in (2.3). Firstly, for all $s ⩾ - log (T_{0} - t_{1} (x_{0}))$ we introduce the following natural functional: $\begin{array}{rcl} E_{0} (w (s), s) & = & \int_{B} (\frac{1}{2} {(\partial_{s} w)}^{2} + \frac{1}{2} (| \nabla w |^{2} - {(y \cdot \nabla w)}^{2}) + \frac{p_{c} + 1}{{(p_{c} - 1)}^{2}} w^{2} - \frac{| w |^{p_{c} + 1}}{p_{c} + 1}) d y \\ (3.1) & - e^{\frac{- 2 (p_{c} + 1) s}{p_{c} - 1}} \int_{B} F (e^{\frac{2 s}{p_{c} - 1}} w) d y . \end{array}$ Moreover, for all $s ⩾ - log (T_{0} - t_{1} (x_{0}))$ , we define the functional $\begin{matrix} (3.2) & H_{0} (w (s), s) = E_{0} (w (s), s) + \frac{1}{s^{\frac{a - 2}{4}}} . \end{matrix}$ We derive that the functional $H_{0} (w (s), s)$ is a decreasing functional of time for Eq. (1.7), provided that s large enough. Let us first control the time derivative of the functional $E_{0} (w (s), s)$ in the following lemma:

Lemma 3.1.
For all $s ⩾ max (- log T^{} (x_{0}), 1)$ , we have the following inequality* $\begin{matrix} (3.3) & \frac{d}{d s} E_{0} (w (s), s) = - \int_{\partial B} {(\partial_{s} w)}^{2} d σ + Σ_{4} (s), \end{matrix}$ where $Σ_{4} (s)$ satisfies $\begin{matrix} (3.4) & Σ_{4} (s) ⩽ \frac{C}{s^{a}} \int_{B} | w |^{p_{c} + 1} d y + C e^{- s} . \end{matrix}$
Proof.
Multiplying (1.7) by $\partial_{s} w$ and integrating over B, we obtain (3.3) where $\begin{matrix} (3.5) & Σ_{4} (s) = \frac{2 (p_{c} + 1)}{p_{c} - 1} e^{- \frac{2 (p_{c} + 1) s}{p_{c} - 1}} \int_{B} F (e^{\frac{2 s}{p_{c} - 1}} w) d y - \frac{2}{p_{c} - 1} e^{- \frac{2 p_{c} s}{p_{c} - 1}} \int_{B} w f (e^{\frac{2 s}{p_{c} - 1}} w) d y . \end{matrix}$ According to (2.15) and (3.5), we get the desired estimate in (3.4), which end the proof of Lemma 3.1. □

With Lemma 3.1 and Theorem 1.1′ we are in position to prove that $H_{0} (w (s), s)$ is a Lyapunov functional of Eq. (1.7), provided that s is large enough. Proposition 3.2.
Consider u a solution of ( 1.1 ) with blow-up graph $Γ : {x \mapsto T (x)}$ and $x_{0}$ is a non-characteristic point. Then, there exists $t_{2} (x_{0}) \in [t_{1} (x_{0}), T (x_{0}))$ such that, for all $T_{0} \in (t_{2} (x_{0}), T (x_{0})]$ , for all $s ⩾ - log (T_{0} - t_{2} (x_{0}))$ and $x \in R^{N}$ such that $| x - x_{0} | ⩽ \frac{e^{- s}}{δ_{0} (x_{0})}$ , we have $\begin{matrix} (3.6) & H_{0} (w (s + 1), s + 1) - H_{0} (w (s), s) ⩽ - \int_{s}^{s + 1} \int_{\partial B} {(\partial_{s} w (σ, τ))}^{2} d σ d τ . \end{matrix}$ Moreover, there exists $S_{6} ⩾ S_{4}$ such that, for all $s ⩾ max (- log (T_{0} - t_{2} (x_{0})), S_{6})$ , we have: $\begin{matrix} H_{0} (w (s), s) ⩾ 0 . \end{matrix}$
Proof.
We apply the polynomial space–time estimate (2.80) in the particular case where $b = \frac{a}{2} > 1$ and Lemma 3.1 to get, for all $s ⩾ - log (T_{0} - t_{1} (x_{0}))$ $\begin{matrix} (3.7) & E_{0} (w (s + 1), s + 1) - E_{0} (w (s), s) ⩽ - \int_{s}^{s + 1} \int_{\partial B} {(\partial_{s} w (σ, τ))}^{2} d σ d τ + \frac{C}{s^{\frac{a}{2}}} . \end{matrix}$ Then we write, for all $s ⩾ - log (T_{0} - t_{1} (x_{0}))$ $\begin{array}{rcl} H_{0} (w (s + 1), s + 1) - H_{0} (w (s), s) & ⩽ & - \int_{s}^{s + 1} \int_{\partial B} {(\partial_{s} w (σ, τ))}^{2} d σ d τ \\ (3.8) & + \frac{C}{s^{\frac{a}{2}}} + \frac{1}{{(s + 1)}^{\frac{a - 2}{4}}} - \frac{1}{s^{\frac{a - 2}{4}}} . \end{array}$ For all $s ⩾ 1$ , by the mean value theorem to the function $x \mapsto \frac{1}{x^{\frac{a + 2}{4}}}$ , between s and $s + 1$ , so we can say that there exists a constant $γ = γ (s) \in (0, 1)$ such that $\begin{matrix} (3.9) & \frac{1}{{(s + 1)}^{\frac{a - 2}{4}}} - \frac{1}{s^{\frac{a - 2}{4}}} = \frac{2 - a}{4 {(s + γ)}^{\frac{a + 2}{4}}} < \frac{2 - a}{4 {(s + 1)}^{\frac{a + 2}{4}}} ⩽ \frac{2 - a}{2^{\frac{a + 10}{4}}} \frac{1}{s^{\frac{a + 2}{4}}} . \end{matrix}$ Finally, by exploiting (3.8) and the inequality (3.9), we can choose $S_{5} > S_{4}$ large enough so that we get (3.6), where $\begin{matrix} (3.10) & t_{2} (x_{0}) = max (T (x_{0}) - e^{- S_{5}}, 0) . \end{matrix}$ To end the proof of the last point of Proposition 3.2, we refer the reader to [25]. This concludes the proof of Proposition 3.2. □

3.2. Proof of Theorem 1.3

In this subsection, we prove Theorem 1.3. Note that the lower bound follows from the finite speed of propagation and the well-posedness in $H^{1} \times L^{2}$ . For a detailed argument in the similar case of Eq. (1.7), see Lemma 3.1, p. 1136 in [25]. Let us first use Proposition 4 and the averaging technique of [25] and [26] to get the following bounds:

Corollary 3.3.

For all $s ⩾ max (- log (T_{0} - t_{2} (x_{0})), S_{6})$ , it holds that $\begin{array}{l} - C ⩽ E_{0} (w (s), s) ⩽ M_{0}, \\ \int_{s}^{s + 1} \int_{\partial B} {(\partial_{s} w (σ, τ))}^{2} d τ ⩽ M_{0}, \\ \int_{s}^{s + 1} \int_{B} {(\partial_{s} w (y, τ) - λ (τ, s) w (y, τ))}^{2} d y d τ ⩽ C M_{0}, \end{array}$ where $0 ⩽ λ (τ, s) ⩽ C$ .

Proof of Theorem 1.3.

The proof is similar to the one in the unperturbed case treated by Merle and Zaag in [25] and [26] and also used by Hamza and Zaag in [14,15] and Hamza and Saidi in [12] and [13]. To be accurate and concise in our results, there is an analogy between the exponential smallness exploited in [14] by Hamza and Zaag and the polynomial smallness used here. The unique difference lies in the treatment of the perturbed term which is treated by Hamza and Saidi [12] and [13]. This concludes the proof of Theorem 1.3. □

Footnotes

Acknowledgements

The author would like thank the reviewers for their valuable comments which undoubtedly helped us to improve the presentation of our results. The author is partially supported by the ERC Advanced Grant no. 291214, BLOWDISOL during his visit to LAGA, Université Paris 13 in 2014.

Some identity related to the Pohozaev multiplier

In this appendix, for all $s ⩾ max (- log T^{*} (x), 1)$ , we evaluate the term $\begin{matrix} (A.1) & L (s) = \int_{B} (y \cdot \nabla w) (div (\nabla w - (y \cdot \nabla w) y)) ψ (y, s) d y, \end{matrix}$ where $\begin{matrix} (A.2) & ψ (y, s) = {(1 - | y |^{2})}^{1 + \frac{1}{s^{b}}} (1 - log (1 - | y |^{2})) . \end{matrix}$ More precisely, we prove the following identity:

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The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption

Abstract

Keywords

1. Introduction

Corollary 2.6 (Estimate on N ( w ( s ) , s ) ).

3.1. Existence of a Lyapunov functional for Eq. (1.7) and a blow-up criterion

Footnotes

Acknowledgements

Some identity related to the Pohozaev multiplier

References

Corollary 2.6 (Estimate on $N (w (s), s)$ ).