Mohamed Ali Hamza

Hatem Zaag

The blow-up rate for a non-scaling invariant semilinear wave equations in higher dimensions

Keywords: MSC 2010 Classification: 35L05, 35B44, 35L71, 35L67, 35B40 Semilinear wave equation, blow-up, non scaling invariance, higher dimensions

We consider the semilinear wave equation

. Our goal here is to extend this result to higher dimensions.

Introduction 1.Motivation of the problem

This paper is devoted to the study of blow-up solutions for the following semilinear wave equation:

   ∂ 2 t u = ∆u + f (u), (x, t) ∈ R N × [0, T ), u(x, 0) = u 0 (x) ∈ H 1 loc,u (R N ), ∂ t u(x, 0) = u 1 (x) ∈ L 2 loc,u (R N ), (1.1)
where u(t) : x ∈ R N → u(x, t) ∈ R with focusing nonlinearity f defined by:

f (u) = |u| p-1 u log a (2 + u 2 ), p > 1, a ∈ R. (1.2) 1
The spaces L 2 loc,u (R N ) and H 1 loc,u (R N ) are defined by

L 2 loc,u (R N ) = {u : R N → R/ sup d∈R N ( |x-d|≤1 |u(x)| 2 dx) < +∞},
and

H 1 loc,u (R N ) = {u ∈ L 2 loc,u (R N ), |∇u| ∈ L 2 loc,u (R N )}.
We assume in addition that p > 1 and if N ≥ 2, we further assume that

p < p c ≡ 1 + 4 N -1 . (1.3)
When a = 0, the nonlinearity in (1.2) is not homogeneous, which means that equation (1.1) is not scale invariant. This is precisely our challenge in this paper, particularly in higher dimensions, since we handled the one dimensional case in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF].

Semilinear wave equations with a nonlinearity showing a logarithmic factor have been introduced in various nonlinear physical models, for instance in the context of nuclear physics, wave mechanics, optics, geophysics etc ... see e.g. [START_REF] Bia Lynicki-Birula | Wave equations with logarithmic nonlinearities[END_REF]6].

The defocusing case has been studied in the mathematical literature and the first results are due to Tao [START_REF] Tao | Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data[END_REF] where the author proved a global well-posedness and scattering result for the three dimensional nonlinear wave equation ∂ 2 t u = ∆u -|u| 4 u log(2 + u 2 ), in the radial case. See also the work of Shih [START_REF] Shih | Some results on scattering for log-subcritical and log-supercritical nonlinear wave equations[END_REF], where the method is refined to treat ∂ 2 t u = ∆u -|u| 4 u log c (2 + u 2 ), for any c ∈ (0, 4 3 ). Later, Roy extends in [START_REF] Roy | Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation[END_REF] the results (global well-posedness and scattering) to solutions of the log-log-supercritical equation ∂ 2 t u = ∆u -|u| 4 u log c log(10 + u 2 ) , for c small, without any radial assumption. Hoping to extend the validity of our argument to the conformal case (p = p c ) in some forthcoming work, we may see the case a > 0 of (1.2) as a further step in the understanding of blow-up dynamics in the superconformal case related to equation (1.8) below.

Let us mention that the blow-up question for the semilinear heat equation ∂ t u = ∆u + |u| p-1 u log a (2 + u 2 ) is studied by Duong-Nguyen-Zaag in [START_REF] Duong | Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation[END_REF]. More precisely, they construct for this equation a solution which blows up in finite time T , only at one blow-up point a, according to the following asymptotic dynamics:

u(x, t) ∼ φ(t) 1 + (p -1)|x -a| 2 4p(T -t)| log(T -t)| -1 p-1 , as t → T, (1.4)
where φ(t) is is the unique positive solution of the ODE φ = |φ| p-1 φ log a (2 + φ 2 ), lim t→T φ(t) = +∞.

(1.5)

Given that we have the same expression in the pure power nonlinearity case (g(u) = |u| p-1 u) with φ(t) replaced by κ(T -t) -1 p-1 (see [START_REF] Bricmont | Universality in blow-up for nonlinear heat equations[END_REF]), we see that the effect of the nonlinearity is all encapsulated in the ODE (1.5).

Equation (1.1) is well-posed in H 1 loc,u × L 2 loc,u . This follows from the finite speed of propagation and the well-posedness in H 1 (R N ) × L 2 (R N ). The existence of blow-up solutions u(t) of (1.1) follows from ODE techniques or the energy-based blow-up criterion by Levine [START_REF] Levine | Instability and nonexistence of global solutions to nonlinear wave equations of the form P u tt = -Au + F (u)[END_REF] (see also [START_REF] Levine | Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy[END_REF][START_REF] Serrin | Existence for a nonlinear wave equation with damping and source terms[END_REF][START_REF] Todorova | Cauchy problem for a non linear wave equation with non linear damping and source terms[END_REF]). More blow-up results can be found in Caffarelli and Friedman [START_REF] Caffarelli | Differentiability of the blow-up curve for onedimensional nonlinear wave equations[END_REF][START_REF]The blow-up boundary for nonlinear wave equations[END_REF], Kichenassamy and Littman [START_REF] Kichenassamy | Blow-up surfaces for nonlinear wave equations[END_REF][START_REF]Blow-up surfaces for nonlinear wave equations[END_REF]. Numerical simulations of blow-up are given by Bizoń et al. (see [START_REF] Bizoń | Threshold behavior for nonlinear wave equations[END_REF][START_REF] Bizoń | Self-similar solutions of the cubic wave equation[END_REF][START_REF] Bizoń | On blowup for semilinear wave equations with a focusing nonlinearity[END_REF][START_REF] Bizoń | Universality of global dynamics for the cubic wave equation[END_REF]).

If u is an arbitrary blow-up solution of (1.1), we define a 1-Lipschitz curve Γ = {(x, T (x))} such that the domain of definition of u is written as

D = {(x, t) | t < T (x)}.
(1.6)

The justification of this fact can be found in Alinhac [START_REF] Alinhac | Blowup for nonlinear hyperbolic equations[END_REF], where such a set is referred to as the maximal influence domain. Let us motivate that particular form, and refer the interested reader to [START_REF] Alinhac | Blowup for nonlinear hyperbolic equations[END_REF] for details.

Recall first that equation (1.1) is well-posed in H 1 × L 2 (R N ), i.e. globally in space (see for example Shatah and Struwe [START_REF] Shatah | Geometric wave equations[END_REF]). Using the finite speed of propagation and a cut-off technique, we then solve the solution locally in space, as Azaiez, Masmoudi and Zaag do in [START_REF] Azaiez | Partial differential equations arising from physic and geometry. A volume in memory of Abbas Bahri[END_REF] in a similar setting. More precisely, for any x 0 ∈ R N , we derive a solution which is defined in K x 0 ,t 0 , the backward light cone defined by

K x 0 ,t 0 = {(ξ, τ ) ∈ R N × [0, t 0 ) | |ξ -x 0 | < t 0 -τ }.
Denoting by T (x 0 ) the supremum of such a t 0 , we fall into one of these two situations: -either for some x 1 ∈ R N , T (x 1 ) = +∞, and in that case, the solution if defined for all x ∈ R N and t ≥ 0. In other words, T (x 0 ) = +∞, for any x 0 ∈ R N ; -or, for all x 1 ∈ R N , T (x 1 ) < +∞, and this is the case we are interested in. In particular, the domain of definition of u is the union of backward light cones of the type K x 0 ,t 0 , and such a union is clearly of the form (1.6) with x → T (x) is 1-Lipschitz. See also Sasaki [START_REF] Sasaki | Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity[END_REF] for a related definition of the domain of definition in some similar setting.

The time T = inf x∈R N T (x) and Γ are called the blow-up time and the blow-up graph of u. A point x 0 is non characteristic if there are δ 0 ∈ (0, 1) and t 0 < T (x 0 ) such that u is defined on

C x 0 ,T (x 0 ),δ 0 ∩ {t ≥ t 0 } (1.7)
where

C x, t, δ = {(x, t) | t < t -δ|x -x|}. If not, x 0 is said to be characteristic.
In this paper, we study the blow-up rate of any singular solution of (1.1). Before going on, it is necessary to mention that the blow-up rate in the case with pure power nonlinearity

∂ 2 t u = ∆u + |u| p-1 u, (x, t) ∈ R N × [0, T ), (1.8)
was studied by Merle and Zaag in [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF][START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF][START_REF]Determination of the blow-up rate for a critical semilinear wave equation[END_REF]. More precisely, they proved that if u is a solution of (1.8) with blow-up graph Γ : {x → T (x)} and x 0 is a non-characteristic point, then, for all t ∈

[ 3T (x 0 ) 4 , T (x 0 )], 0 < ε 0 (p) ≤ (T (x 0 ) -t) 2 p-1 u(t) L 2 (B(x 0 ,T (x 0 )-t)) (T (x 0 ) -t) N 2 (1.9) 3 +(T (x 0 ) -t) 2 p-1 +1 ∂ t u(t) L 2 (B(x 0 ,T (x 0 )-t)) (T (x 0 ) -t) N 2 + ∂ x u(t) L 2 (B(x 0 ,T (x 0 )-t)) (T (x 0 ) -t) N 2 ≤ K 0 ,
where the constant K 0 depends only on p and on an upper bound on T (x 0 ), 1/T (x 0 ), δ 0 (x 0 ), together with the norm of initial data in

H 1 loc,u (R N ) × L 2 loc,u (R N ).
Namely, the blow-up rate of any singular solution of (1.8) is given by the solution of the associated ODE u = |u| p-1 u. Note that this result about the blow-up rate is valid in the subconformal and conformal case (1 < p ≤ p c ).

In a series of papers, Merle and Zaag [START_REF]Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension[END_REF][START_REF]Openness of the set of non characteristic points and regularity of the blowup curve for the 1 d semilinear wave equation[END_REF][START_REF]Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension[END_REF][START_REF]Isolatedness of characteristic points for a semilinear wave equation in one space dimension[END_REF] (see also Côte and Zaag [START_REF] Côte | Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension[END_REF]) give a full picture of blow-up for solutions of equation (1.8) 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 no longer holds because there is no classification of selfsimilar solutions of equation (1.8) in the energy space. However, in the radial case outside the origin, Merle and Zaag reduce the blow up description to the one-dimensional case with perturbation and obtain the same results as for N = 1 (see [START_REF]Blow-up behavior outside the origin for a semilinear wave equation in the radial case[END_REF] and also the extension by Hamza and Zaag in [START_REF]Blow-up behavior for the klein-gordon and other perturbed semilinear wave equations[END_REF] to the Klein-Gordon equation and other damped lower-order perturbations of equation (1.8)). Later, Merle and Zaag could address the higher dimensional case in the subconformal case and prove the stability of the explicit selfsimilar solution with respect to the blow-up point and initial data (see [START_REF]On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations[END_REF][START_REF]Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions[END_REF]). 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) -2 p-1 in the light cone with respect to small perturbations in initial data, in a stronger topology (see [START_REF] Donninger | Stable self-similar blow up for energy subcritical wave equations[END_REF][START_REF]Stable blow up dynamics for energy supercritical wave equations[END_REF][START_REF]On blowup in supercritical wave equations[END_REF][START_REF]Stable blowup for wave equations in odd space dimensions[END_REF]). Their approach is based in particular on a good understanding of the spectral properties of the linearized operator in selfsimilar variables, operator which is not self-adjoint. Later, thanks to suitable Strichartz estimates for the critical wave equation in similarity variables, Donninger proved in [START_REF] Donninger | Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation[END_REF] the stability of the solution of the ODE with respect to small perturbations in initial data, in the energy space. Let us also mention that Killip, Stoval and Vişan proved in [START_REF] Killip | Blowup behaviour for the nonlinear Klein-Gordon equation[END_REF] 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 [START_REF]Blow-up results for semilinear wave equations in the super-conformal case[END_REF].

In [START_REF] Hamza | A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case[END_REF][START_REF]A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations[END_REF], using a highly non-trivial perturbative method, we could obtain the blow-up rate for the Klein-Gordon equation and more generally, for equation

∂ 2 t u = ∆u + |u| p-1 u + f (u) + g(∂ t u), (x, t) ∈ R N × [0, T ), (1.10)
under the assumptions |f (u)| ≤ M (1 + |u| q ) and |g(v)| ≤ M (1 + |v|), for some M > 0 and q < p ≤ N +3 N -1 . In fact, we proved a similar result to (1.9), valid in the subconformal and conformal case. Let us also mention that in [START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption[END_REF][START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations[END_REF][START_REF]The blow-up rate for strongly perturbed semilinear wave equations in the conformal case[END_REF], the results obtained in [START_REF] Hamza | A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case[END_REF][START_REF]A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations[END_REF] were extended by Hamza and Saidi to the strongly perturbed equation (1.10) with |f (u)| ≤ M (1 + |u| p log -a (2 + u 2 )), for some a > 1, though keeping the same condition in g. Very recently, Azaiez and Zaag derived in [START_REF] Azaiez | Classification of the blow-up behavior for a semilinear wave equation with nonconstant degenerate coefficients[END_REF] the blow-up rate for equations of the type

∂ 2 t u = a(x)(∂ 2 x u + N -1 x ∂ x u) + b(x)|u| p-1 u + f (u) + g(x, t, ∂ x u, ∂ t u)
where |f (u)| ≤ M (1 + |u| q ) with q < p, |g(x, t, v, z)| ≤ M (1 + |v| a(x) + |z|), for some M > 0, and a(x) is typically |x| α with α enjoying a infinite number of values converging to 2.

In the previous works [START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption[END_REF][START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations[END_REF][START_REF]The blow-up rate for strongly perturbed semilinear wave equations in the conformal case[END_REF][START_REF] Hamza | A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case[END_REF][START_REF]A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations[END_REF], we consider a class of perturbed equations where the nonlinear term is equivalent to the pure power |u| p-1 u and we obtain the estimate (1.9). This is due to the fact that the dynamics are governed by the ODE equation: u = |u| p-1 u and not influenced by perturbative terms. Furthermore, our proof remains (non trivially) perturbative with respect to the homogeneous PDE (1.8), which is scale invariant. This leaves unanswered an interesting question: Is the scale invariance property crucial in deriving the blow-up rate?

In fact we had the impression that the answer was "yes", since the scaling invariance induces in similarity variables a PDE which is autonomous in the unperturbed case (1.8), and asymptotically autonomous in the perturbed case (1.10).

In this paper we prove that the answer is "no" from the example of the PDE (1.1) with the non homogeneous nonlinearity (1.2). In fact, our situation is different from (1.8) and (1.10), in the sense that the term |u| p-1 u log a (2 + u 2 ) is playing a fundamental role in the dynamics of the blow-up solution of (1.1). More precisely, we obtain an analogous result to (1.9) but with a logarithmic correction as shown in (1.27) below. In fact, the blow-up rate is given by the solution of the following ordinary differential equation:

v

T (t) = |v T (t)| p-1 v T (t) log a v 2 T (t) + 2 , v(T ) = ∞, (1.11)
which satisfies

v T (t) ∼ κ a ψ T (t), as t → T, where κ a = 2 1-2a (p + 1) (p -1) 2-a 1 p-1 , (1.12)
and

ψ T (t) = (T -t) -2 p-1 (-log(T -t)) -a p-1 (1.13)
(see Lemma A.2 in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF]).

Finally, let us remark that, following the analysis of Hamza in Saidi in [START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations[END_REF], our result holds for this more general equation

∂ 2 t u = ∆u + f (u) + g(x, t, u, ∂ t u, ∇u), (x, t) ∈ R N × [0, T ), (1.14)
where f : R → R and g : R 2N +3 → R are C 1 functions which satisfy the following conditions:

|g(x, t, u, v, w)| ≤ M 0 (1 + |u| p+1 2 + |v| + |w|), (1.15)
and

u 0 f (y)dy - u f ux) p + 1 + 2au f (u) (p + 1) 2 log(2 + u 2 ) ≤ M 0 1 + |u| p+1 log b-1 (2 + u 2 ) , (1.16)
for some M 0 > 0, p > 1, a ∈ R and b < a. In fact, under these assumptions, we can construct a suitable Lyapunov functional for the equation (1.14) under hypothesis (1.15) and (1.16). Then, we can prove a similar result to (1.27) with the necessary modifications. However, In order to keep our analysis clear, we restrict our analysis to the problem related to equation (1.1).

Strategy of the proof

Going back to the equation under study in this paper (see (1.1) and (1.2)), we introduce the following similarity variables, defined for all x 0 ∈ R N , T 0 such that 0 < T 0 ≤ T (x 0 ) by:

y = x -x 0 T 0 -t , s = -log(T 0 -t), u(x, t) = ψ T 0 (t)w x 0 ,T 0 (y, s).
(1.17)

One may think that it would be more natural to replace ψ T 0 (t) by v T 0 (t) in this definition, since the latter is an exact solution of the ODE (1.11), unlike the former. That might be a good idea, however, as v T 0 (t) has no explicit expression, the calculations will immediately become too complicated. For that reason, we preferred to replace the non-explicit v T 0 (t) by its explicit equivalent ψ T 0 (t) in (1.13). The fact that the latter is only an approximate solution and not an exact solution of (1.11) will have no effect on our analysis. From (1.1), the function w x 0 ,T 0 (we write w for simplicity) satisfies the following equation for all y ∈ B and s ≥ max(-log T 0 , 1), where B ≡ B(0, 1) stands for the unit ball of R N and throughout the paper:

∂ 2 s w = 1 ρ div (ρ∇w -ρ(y.∇w)y) + 2a (p -1)s y.∇w - 2p + 2 (p -1) 2 w + γ(s)w - p + 3 p -1 - 2a (p -1)s ∂ s w -2y.∇∂ s w + e -2ps p-1 s a p-1 f (φ(s)w), (1.18) with ρ(y) = (1 -|y| 2 ) α , α = 2 p -1 - N -1 2 > 0, (1.19)
γ(s) = a(p + 5) (p -1) 2 s - a(p + a -1) (p -1) 2 s 2 , (1.20) and φ(s) = e 2s p-1 s -a p-1 . (1.21)
In the new set of variables (y, s), the behavior of u as t → T 0 is equivalent to the behavior of w as s → +∞. Moreover, if T 0 = T (x 0 ), then we simply write w x 0 instead of w x 0 ,T (x 0 ) .

The equation (1.18) will be studied in the Hilbert space

H H = (w 1 , w 2 ), | B w 2 1 + |∇w 1 | 2 -|y.∇w 1 | 2 + w 2 2 dy < +∞ .
As in the pure power case (1.8) and the perturbed case (1.10), the construction of a Lyapunov functional in similarity variables was the starting point of our strategy. In the present case (1.1), we adopt the same strategy. We were successful in implementing that in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF], however, only in one space dimension. Indeed, our method in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF] breaks down in higher dimensions. Let us briefly explain in the following the method used in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF] and how it breaks down in higher dimensions, giving sense to the present work.

The first step in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF] consists in the introduction of a functional associated to equation (1.18) which satisfies the following differential inequality:

d ds h(s) ≤ -α B (∂ s w) 2 ρ(y) 1 -|y| 2 dy + C s h(s),
where α is defined in (1. [START_REF]Stable blowup for wave equations in odd space dimensions[END_REF]) and w is the solution of (1.18). Thanks to the abovementioned functional, we easily derived a polynomial (in s) bound for the

H 1 × L 2 (B)
norm of the solution of (1.18) (in space-time averages). More precisely, we obtain the estimates (2.1), (2.2) and (2.3) below. Let us recall that the nonlinear term f (u) given by (1.2) is not a pure power. This is why the strategy used to remove the time averages in the case of pure power in Merle and Zaag [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF][START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF][START_REF]Determination of the blow-up rate for a critical semilinear wave equation[END_REF], and naturally implemented in our previous papers [START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption[END_REF][START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations[END_REF][START_REF]The blow-up rate for strongly perturbed semilinear wave equations in the conformal case[END_REF][START_REF] Hamza | A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case[END_REF][START_REF]A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations[END_REF] in the perturbative cases, breaks down in higher dimensions. Indeed, this method is somehow based on some interpolation results in Sobolev spaces, and some critical Gagliardo-Nirenberg estimates. However, in one dimension, the strategy used to remove the time average works since it is based on the embedding

H 1 (R × [-log T, +∞)) → L q (R × [-log T, +∞)), for any q > 1.
Using the polynomial (in s) bound for the H 1 loc,u (R)-norm of the solution of (1.18) and the embedding H 1 (R) → L ∞ (R), we derive a Lyapunov functional for equation (1.18) in one space dimension, which is a crucial step to obtain the optimal estimate. Since the embedding of H 1 into L q for any q > 1 is specific to dimension 1 + 1 and doesn't hold in dimension N + 1, the higher dimensional case requires new ideas, which we explain in the following.

First, we recall in (2.1), (2.2) and (2.3) the rough polynomial (in s) bound (in spacetime averages) on the solution near any non characteristic point in similarity variables, was proved and stated in any dimensions in the subconformal case (p < p c ). Moreover, we can somehow reduce to the pure power case, up to an ε perturbation, as we write in the elementary estimates on the nonlinear term given below in (A.4), (A.7) and (A.8). In fact, by exploiting these estimates, we prove an improved version of the estimates (2.1) and (2.2), where we remove the time average. Then, using these new estimates, the embedding of

H 1 (R N ) in L 2 * (R N ) if N ≥ 3 and in L q (R N ) for any q ≥ 2 if N = 2,
together with the structure of the nonlinear term, we end up with the construction of a functional g(s) which satisfies the following differential inequality:

d ds g(s) ≤ -α B (∂ s w) 2 ρ(y) 1 -|y| 2 dy + C s 3 2 g(s) + C s 7 4 , (1.22)
where α is defined in (1.19) and w is the solution of (1.18). Naturally, by (1.22), we easily derive a Lyapunov functional for equation (1.18), valid in any dimensions in the subconformal case, and this is our main contribution in this work. With this Lyapunov functional at hand, the adaptation of the interpolation strategy from our previous papers works straightforwardly.

Statement of the results

To state our main result, we start by introducing the following functionals:

E(w(s), s) = B 1 2 (∂ s w) 2 + 1 2 (|∇w| 2 -(y.∇w) 2 ) + p + 1 (p -1) 2 w 2 -e -2(p+1)s p-1 s 2a p-1 F (φw) ρ(y)dy, (1.23)
L 0 (w(s), s) = E(w(s), s) - 1 s √ s B ∂ s wwρ(y)dy, (1.24)
where

F (u) = u 0 f (v)dv = u 0 |v| p-1 v log a (v 2 + 2)dv. (1.25)
Moreover, for all s ≥ max(1, -log T 0 ), we define the functional

L(w(s), s) = exp p + 3 √ s L 0 (w(s), s) + θs -3 4 , (1.26)
where θ is a sufficiently large constant that will be determined later. We will show that the functional L(w(s), s) is a decreasing functional of time for equation (1.18), provided that s is large enough. Clearly, by (1.24) and (1.26), the functional L(w(s), s) is a small perturbation of the "natural" energy E(w(s), s).

Here is the statement of our main theorem in this paper.

Theorem 1. Let N ≥ 2, 1 < p < 1 + 4 N -1 and a ∈ R. Let f defined by: f (u) = |u| p-1 u log a (2 + u 2 ). Consider u a solution of (1.1) with blow-up graph Γ : {x → T (x)}, and x 0 a non characteristic point. Then there exists t 1 (x 0 ) ∈ [0, T (x 0 )) such that, for all T 0 ∈ (t 1 (x 0 ), T (x 0 )], for all s ≥ -log(T 0 -t 1 (x 0 )), we have

L(w(s + 1), s + 1) -L(w(s), s) ≤ -α s+1 s B (∂ s w) 2 ρ(y) 1 -|y| 2 dydτ,
where w = w x 0 ,T 0 is defined in (1.17).

Remark 1.1. Since the existence of a Lyapunov functional in similarity variables is far from being trivial and represents the crucial step in this paper, we choose to state it first in our paper, and to give it the status of a "first theorem".

Remark 1.2. As we mentioned before, we can construct a suitable Lyapunov functional in similarity variables for the more general equation (1.14) under hypotheses (1.15) and (1.16), then show that Theorem 1 holds true for this functional.

Remark 1.3. Let us note that our method breaks down in the case of a characteristic point, since in the construction of the Lyapunov functional in similarity variables, we use a covering technique in our argument which is not available at a characteristic point. At this moment, we do not know whether Theorem 1 continues to hold if x 0 is a characteristic point.

As we said earlier, the existence of this Lyapunov functional L(w(s), s) together with a blow-up criterion for equation (1.18) make a crucial step in the derivation of the blow-up rate for equation (1.1). Indeed, with the functional L(w(s), s) and some more work, we are able to adapt the analysis performed in [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF][START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF][START_REF]Determination of the blow-up rate for a critical semilinear wave equation[END_REF] for equation (1.8) and obtain the following result:

Theorem 2. (Blow-up rate for equation (1.1)). Let N ≥ 2, 1 < p < 1 + 4
N -1 and a ∈ R. Consider u a solution of (1.1), where f (u) = |u| p-1 u log a (2 + u 2 ), with blow-up graph Γ : {x → T (x)} and x 0 a non characteristic point. Then, there exists S 2 large enough such that i) For all s ≥ s 2

(x 0 ) = max( S 2 , -log T (x 0 ) 4 ), 0 < ε 0 ≤ w x 0 (s) H 1 (B) + ∂ s w x 0 (s) L 2 (B) ≤ K,
where w x 0 = w x 0 ,T (x 0 ) is defined in (1.17), for some ε 0 = ε 0 (N, p, a) and

K = K(N, p, a, T (x 0 ), t 2 (x 0 ), (u(t 2 (x 0 )), ∂ t u(t 2 (x 0 ))) H 1 ×L 2 (B(x 0 , T (x 0 )-t 2 (x 0 ) δ 0 (x 0 )
)) ), ψ T (x 0 ) (t) is defined in (1.13) and δ 0 (x 0 ) is given by (1.7).

. ii) For all t ∈ [t 2 (x 0 ), T (x 0 )), where t 2 (x 0 ) = T (x 0 ) -e -s 2 (x 0 ) , we have

0 < ε 0 ≤ 1 ψ T (x 0 ) (t) u(t) L 2 (B(x 0 ,T (x 0 )-t)) (T (x 0 ) -t) N 2 (1.27) + T (x 0 ) -t ψ T (x 0 ) (t) ∂ t u(t) L 2 (B(x 0 ,T (x 0 )-t)) (T (x 0 ) -t) N 2 + ∇u(t) L 2 (B(x 0 ,T (x 0 )-t)) (T (x 0 ) -t) N 2 ≤ K,
where ε 0 and K are introduced in item i).

Remark 1.4. As we said before, we are able to obtain a similar result to (1.27) for the more general equation (1.14) under hypotheses (1.15) and (1.16).

Remark 1.5. Both for the construction of the Lyapunov functional and the derivation of the bounds in this theorem, our method breaks down in the conformal case, even when a < 0, and we are not able to obtain the sharp estimate as in the case of a pure power nonlinearity (1.8) treated in [START_REF]Determination of the blow-up rate for a critical semilinear wave equation[END_REF].

Remark 1.6. Since we crucially need a covering technique in the argument of the construction of the Lyapunov functional, our method breaks down too in the case of a characteristic point and we are not able to obtain the sharp estimate as in the unperturbed case (1.8).

Remark 1.7. As in [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] in the pure power nonlinearity case (1.8), the proof of Theorem 2 relies on four ideas: the existence of a Lyapunov functional, interpolation in Sobolev spaces, some critical Gagliardo-Nirenberg estimates and a covering technique adapted to the geometric shape of the blow-up surface. As we said before, the first point where we construct a Lyapunov functional in similarity variables is far from being trivial and represent a crucial step. Consequently, we have chosen to present our main contribution as Theorem (1) and we write a detailed proof. However, for the other three points, the adaption of the proof of [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] given in the pure power nonlinearity case (1.8) is straightforward except for a key argument, where we bound the nonlinear term e -2(p+1)s p-1 s 2a p-1 F (φ(s)w). Therefore, in order to avoid unnecessary repetition, we prove this step and kindly refer to [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF][START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF][START_REF]Determination of the blow-up rate for a critical semilinear wave equation[END_REF][START_REF] Hamza | A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case[END_REF][START_REF]A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations[END_REF][START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption[END_REF][START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations[END_REF][START_REF]The blow-up rate for strongly perturbed semilinear wave equations in the conformal case[END_REF] for the rest of the proof. This paper is organized as follows: In Section 2, we obtain a polynomial (in s) bound for the H 1 × L 2 (B) norm of the solution (w, ∂ s w). In Section 3, thanks to this result, we prove that the functional L(w(s), s) defined in (1.26) is a Lyapunov functional for equation (1.18). Thus, we get Theorem 1. Finally, applying this last theorem, we prove Theorem 2.

Throughout this paper, C denotes a generic positive constant depending only on p, N and a, which may vary from line to line. In addition, we will use K 1 , K 2 , K 3 ... as a positive constants depending only on p, N, a, δ 0 (x 0 ) and initial data, which may also vary from line to line. We write

f (s) ∼ g(s) to indicate lim s→∞ f (s) g(s) = 1.
2 A polynomial bound for the

H 1 × L 2 (B) norm of solution of equation (1.18)
Let us first recall the rough polynomial space-time estimate of the solution u of (1.1) near any non characteristic point obtained in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF] (see Theorem 1). More precisely, we established the following results:

(Polynomial space-time estimate of solution of (1.18)). There exists q = q(a, p, N ) > 0 such that for any u a solution of (1.1) with blow-up graph Γ : {x → T (x)} and x 0 a non characteristic point. Then, there exists t 0 (x 0 ) ∈ [0, T (x 0 )) such that, for all T 0 ∈ (t 0 (x 0 ), T (x 0 )], for all s ≥ -log(T 0 -t 0 (x 0 )) and x ∈ R N where |x-x 0 | ≤ T 0 -t δ 0 (x 0 ) , we have

s+1 s B |∇w| 2 + (∂ s w) 2 dydτ ≤ K 1 s q , (2.1) 1 s a s+1 s B |w| p+1 log a (2 + φ 2 w 2 )dydτ ≤ K 1 s q , (2.2) B w 2 dy ≤ K 1 s q , (2.3)
where w = w x,T * (x) is defined in (1.17), with

T * (x) = T 0 -δ 0 (x 0 )|x -x 0 | (2.4)
and δ 0 (x 0 ) defined in (1.7). Note that K 1 depends on p, a, N, δ 0 (x 0 ), T (x 0 ), t 0 (x 0 ) and

(u(t 0 (x 0 )), ∂ t u(t 0 (x 0 ))) H 1 ×L 2 (B(x 0 , T (x 0 )-t 0 (x 0 ) δ 0 (x 0 )
)) . Moreover, we have

-K 1 s q ≤ H m 0 (w(s), s) ≤ K 1 s q , (2.5)
where

H m 0 (w(s), s) = E(w(s), s) - m 0 s B w∂ s wρ(y)dy, (2.6)
E(w(s), s) is given by (1.23) and m 0 is a sufficiently large constant.

This section is devoted to deriving a polynomial bound for the H 1 (B) norm. More precisely, this is the aim of this section.

Proposition 2.1. There exists q 1 = q 1 (a, p, N ) > 0 such that for any solution u of (1.1) with blow-up graph Γ : {x → T (x)} and x 0 a non characteristic point, there exists

t 0 (x 0 ) ∈ [0, T (x 0 )) such that for all T 0 ∈ (t 0 (x 0 ), T (x 0 )], s ≥ -log(T 0 -t 0 (x 0 )) and x ∈ R N with |x -x 0 | ≤ T 0 -t δ 0 (x 0 ) , we have w(s) H 1 (B) ≤ K 2 s q 1 , (2.7)
where w = w x,T * (x) is defined in (1.17), with T * (x) given in (2.4) and δ 0 (x 0 ) defined in (1.7). Note that K 2 depends on p, a, δ 0 (x 0 ), T (x 0 ), t 0 (x 0 ) and

(u(t 0 (x 0 )), ∂ t u(t 0 (x 0 ))) H 1 ×L 2 (B(x 0 , T (x 0 )-t 0 (x 0 ) δ 0 (x 0 )
)) .

Remark 2.1. Let us insist on the fact that the strategy of the proof works only in the subconformal case. Obviously, we can also prove that ∂ s w(s) L 2 (B) ≤ K 2 s q 1 . However, since this estimate is not useful in the proof, we have chosen not to include it in the above proposition.

Remark 2.2. By using the Sobolev embedding and the above-mentioned proposition, we can deduce that for all r ∈ [2, 2 * ] where 2 * = 2N N -2 if N ≥ 3, and for all r ∈ [2, ∞) if N = 2: w(s) L r (B) ≤ K 3 s q 1 , for all s ≥ -log(T * (x) -t 0 (x 0 )), (2.8) where K 3 depends also on r.

Let us prove Proposition 2.1 in the following.

Proof of Proposition 2.1: We proceed in 2 steps: -In Step 1, we use the covering technique and interpolation to derive a polynomial estimate related to the L p+3-ε 2

(B) norm of w(s), for any ε ∈ (0, p -1). -In Step 2, using Step 1, a Gagliardo-Nirenberg estimate and estimate (2.5) satisfied by H m 0 (w(s), s) defined in (2.6), we easily conclude the proof of estimate (2.7).

Step 1: Control of w in L p+3-ε 2 (B) We claim the following: Lemma 2.2. For all ε ∈ (0, p -1) and s ≥ -log(T * (x) -t 0 (x 0 )), we have

B |w(y, s)| p+3-ε 2 dy ≤ K 4 (ε)s q .
(2.9)

Proof: Let us recall from the expression of φ = φ(s) defined in (1.21) that we have for all s ≥ max(-log T * (x), 1),

e -2(p+1)s p-1 s 2a p-1 φwf (φw) = 1 s a |w| p+1 log a (2 + φ 2 w 2 ).
(2.10)

Combining (2.10), (A.4), (A.10) and (2.2), we deduce that for all s ≥ -log(T * (x)t 0 (x 0 )),

s+1 s B |w| p+1-ε dydτ ≤ C(ε)K 1 s q . (2.11)
Now, for all s ≥ -log(T * (x) -t 0 (x 0 )), using the mean value theorem, we derive the existence of σ(s)

∈ [s, s + 1] such that B |w(y, σ(s))| p+1-ε dy = s+1 s B |w(y, τ )| p+1-ε dydτ. (2.12)
Let us introduce the following identity for all s ≥ -log(T * (x) -t 0 (x 0 )),

B |w(y, s)| p+3-ε 2 dy = B |w(y, σ(s))| p+3-ε 2 dy + s σ(s) d dτ B |w(y, τ )| p+3-ε 2 dydτ. (2.13)
Combining (2.12), (2.13) and the fact that xy ≤ x 2 + y 2 , for all x ≥ 0, y ≥ 0, we write for all s ≥ -log(T * (x) -t 0 (x 0 )), ≤ 1+x p+1-ε , for all x ≥ 0, ε ∈ (0, p-1), (2.1) and (2.11), we obtain (2.9). This concludes the Lemma 2.2.

Step 2: Control of ∇w in L 2 (B). As in the pure power case, we first use the Gagliardo-Nirenberg inequality in order to obtain the following: Lemma 2.3. There exists ε 0 = ε 0 (p, N ) > 0 such that, for all ε ∈ (0, ε 0 ], for all s ≥ -log(T * (x) -t 0 (x 0 )), we have

B |w(y, s)| p+1+ε dy ≤ K 5 s q B |∇w(y, s)| 2 dy β(ε) + K 5 s 2q , (2.15)
where β = β(p, N, ε) ∈ (0, 1).

Proof: We distinguish two cases:

• First case (N = 2): Let ε > 0 and r = r(p, ε) > p + 1 + ε. By interpolation, we write (2.17) where

β(r, p, ε) = r(p -1 + 3ε) 4r -2(p + 3 -ε)
.

The combination of lim r→∞ β(r, p, ε) = p -1 + 3ε 4 and p-1 4 < 1 implies that there exists ε 0 = ε 0 (p) = 5-p 10 > 0 such that for ε ∈ (0, ε 0 ] we can choose r large enough such that β = β(r, p, ε) ∈ (0, 1). The result (2.15) follows immediately from (2.17), (2.3) and (2.9).

• Second case (N ≥ 3):

Let ε ∈ (0, p -1). Let us write the following Gagliardo-Nirenberg inequality:

η = 1 -p+1+ε 2 * 1 -p+3-ε 2•2 * , β = β(p, N, ε) = p-1+3ε 4 1 -p+3-ε 2•2 * and 1 2 * = N -2 2N .
Observe that the function p → β(p, N, ε) is an increasing function on (1, p c ), hence,

β(p, N, ε) < β(p c , N, ε), ∀p ∈ (1, p c ). (2.19)
Thanks to (2.19), the fact β(p c , N, 0) = 1 and by continuity, we infer that there exists ε 0 = ε 0 (p, N ) > 0 small enough such that for ε ∈ (0, ε 0 ] we have β = β(p, N, ε) ∈ (0, 1). The result (2.15) follows immediately from (2.18), (2.3) and (2.9), which ends the proof of Lemma 2.3. Now, we are ready to give the proof of Proposition 2.1.

Proof of Proposition 2.1: Let us first use the following covering lemma from [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF]: using Proposition 3.3 in that paper with η = 2(p+1) p-1 -N (which is positive), q = 2 and f = ∇u, together with the selfsimilar change of variables (1.17), we write (2.23)

sup {x | |x-x 0 |≤ T 0 -t δ 0 } B |∇w| 2 dy ≤ C(δ 0 ) sup {x | |x-x 0 |≤ T 0 -t δ 0 } B(0,
Thanks to (A.9) and (2.23), we conclude for all s ≥ -log(T * (x) -t 0 (x 0 )),

B |∇w(y, s)| 2 (1 -|y| 2 )ρ(y)dy ≤ K 7 B |w(y, s)| p+ε+1 ρ(y)dy + K 7 s q . (2.24)
According to (2.24) together with Lemma 2.3, we have for all s ≥ -log(T * (x) -t 0 (x 0 )),

B(0, 1 2 ) |∇w(y, s)| 2 dy ≤ C B |∇w(y, s)| 2 (1 -|y| 2 )ρ(y)dy ≤ K 8 s q |∇w(y, s)| 2 dy β + K 8 s 2q . (2.25) Therefore, sup {x | |x-x 0 |≤ T 0 -t δ 0 } B(0, 1 2 ) |∇w(y, s)| 2 dy ≤ K 8 s q   sup |x-x 0 |≤ T 0 -t δ 0 B |∇w(y, s)| 2 dy   β + K 8 s 2q .
(2.26) where β ∈ (0, 1). From (2.20) and (2.26), we see that

sup {x | |x-x 0 |≤ T 0 -t δ 0 } B |∇w(y, s)| 2 dy ≤ K 9 s q   sup |x-x 0 |≤ T 0 -t δ 0 B |∇w(y, s)| 2 dy   β + K 9 s 2q .
(2.27) It suffices to combine (2.27) and the fact that β < 1, to obtain that

sup {x | |x-x 0 |≤ T 0 -t δ 0 } B |∇w(y, s)| 2 dy ≤ K 10 s 2q 1-β . (2.28)
Clearly, by using (2.28) and (2.3), we conclude (2.7), where q 1 = 1 1-β q, which yields the conclusion of Proposition 2.1.

Proof of Theorem 1 and Theorem 2

In this section, we prove Theorem 1 and Theorem 2 here thanks to Proposition 2.1. This section is divided into two parts:

• In subsection 3.1, we state a general version of Theorem 1, uniform for x near x 0 and prove it.

• In subsection 3.2, we prove Theorem 2.

A Lyapunov functional

In this subsection, our aim is to construct a Lyapunov functional for equation (1.18). Note that this functional is far from being trivial and makes our main contribution. More precisely, thanks to the rough estimate obtained in the Proposition 2.1, we derive here that the functional L(w(s), s) defined in (1.26) is a decreasing functional of time for equation (1.18), provided that s is large enough. First, thanks to the additional information obtained in Section 2, we can write this important lemma which plays a key role in our analysis. More precisely, we claim the following:

Lemma 3.1. For all s ≥ -log(T * (x) -t 0 (x 0 )), we have B |w| p+1 log a (2 + φ 2 w 2 ) log(2 + w 2 )ρ(y)dy ≤K 11 s 1 4 B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + K 11 s a+ 1 4 . (3.1)
Remark 3.1. Let us mention that, in the first term on the right-hand side, the choice of the power 1 4 is not optimal. In fact, with the same proof, one can show the same estimate with the power ν, for any ν > 0, instead of the power 1 4 . Let us remark that we can construct a Lyapunov functional, when we have the estimate above for some power ν such that ν ∈ (0, 1) instead of the power 1 4 . Proof: Let ε ∈ (0, 1). By using the inequality log

(2 + z 2 ) ≤ C(ε) + |z| ε 2 , for all z ∈ R, we conclude that B |w| p+1 log a (2 + φ 2 w 2 ) log(2 + w 2 )ρ(y)dy ≤ C B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + B |w| p+1+ε 2 log a (2 + φ 2 w 2 )ρ(y)dy. (3.2)
Furthermore, we apply the interpolation in Lebesgue spaces to get

B |w| p+1+ε 2 log a (2 + φ 2 w 2 )ρ(y)dy ≤ B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy 1-ε B |w| p+1+ε log a (2 + φ 2 w 2 )ρ(y)dy ε . (3.3)
By combining(A.4), (A.9) and the inequality |z| ε ≤ 1+|z| p+1+2ε , for all z ∈ R, we obtain

1 s a B |w| p+1+ε log a (2 + φ 2 w 2 )ρ(y)dy ≤ C + C B |w| p+1+2ε dy. (3.4) Since p < p c = N +3 N -1 , we then choose ε 1 ≤ ε 0 small enough, such that for all ε ∈ (0, ε 1 ] we have p + 1 + 2ε < 2 * where 2 * = 2N N -2 , if N ≥ 3 and 2 * = ∞, if N = 2.
Therefore, estimate (2.8) implies that, for all s ≥ -log(T * (x) -t 0 (x 0 ))

B |w| p+1+2ε dy ≤ (K 3 s q 1 ) p+1+2ε , ∀ε ∈ [0, ε 1 ]. (3.5)
By combining (3.3), (3.4) and (3.5), we deduce that, for all s ≥ -log(T * (x) -t 0 (x 0 )), for all ε ∈ (0,

ε 1 ]. B |w| p+1+ε 2 log a (2 + φ 2 w 2 )ρ(y)dy ≤ K 12 s q 1 (p+1+2ε)ε s εa B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy 1-ε . (3.6)
Thanks to the basic inequality

|X| ν |Y | 1-ν ≤ C|X| + C|Y |, for all X, Y ∈ R, for all ν ∈ (0, 1), we conclude that, for all s ≥ -log(T * (x) -t 0 (x 0 )), for all ε ∈ (0, ε 1 ]. B |w| p+1+ε 2 log a (2 + φ 2 w 2 )ρ(y)dy ≤ K 13 s q 1 (p+1+2ε)ε s a + B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy . (3.7)
We choose ε 2 ∈ (0, ε 1 ], such that q 1 (p + 1 + 2ε 2 )ε 2 < 1 4 . Then, by (3.2) and (3.7), we easily obtain (3.1). This concludes the proof of Lemma 3.1.

Thanks to estimate (3.1), we can improve the estimate related to the control of the time derivative of the functional E(w(s), s). More precisely, we prove the following lemma: Lemma 3.2. There exists S 1 > 0 such that for all s ≥ max(-log(T * (x) -t 0 (x 0 )), S 1 ), we have

d ds E(w(s), s) ≤ - 3α 2 B (∂ s w) 2 ρ(y) 1 -|y| 2 dy + K 14 s a+ 7 4 B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + C s 2 B |∇w| 2 (1 -|y| 2 )ρ(y)dy + C s 2 B w 2 ρ(y)dy + K 14 s 7 4
.

Proof: Multiplying (1.18) by ∂ s w ρ(y) and integrating over B, we obtain

d ds E(w(s), s) = -2α B (∂ s w) 2 ρ(y) 1 -|y| 2 dy (3.9) + a (p + 1)s a+1 B |w| p+1 log a-1 (2 + φ 2 w 2 ) log(2 + φ 2 w 2 ) - 4s p -1 ρ(y)dy χ 1 (s) + 2e -2(p+1)s p-1 p -1 s 2a p-1 B (p + 1)F 2 (φw) - a s F 1 (φw) - a s F 2 (φw) ρ(y)dy χ 2 (s) + γ(s) B w∂ s wρ(y)dy + 2a (p -1)s B (∂ s w) 2 ρ(y)dy χ 3 (s) + 2a (p -1)s B y.∇w∂ s wρ(y)dy χ 4 (s)
,

where F 1 and F 2 are defined by

F 1 (x) = - 2a (p + 1) 2 |x| p+1 log a-1 (2 + x 2 ), (3.10)
and

F 2 (x) = F (x) - xf (x) p + 1 -F 1 (x). (3.11)
Note that, in (3.9) we grouped the main terms together. In fact, it is easy to control the terms χ 2 (s), χ 3 (s) and χ 4 (s). However, the control of the term χ 1 (s) needs the use of the additional information obtained in Lemma 3.1. More precisely, for all s ≥ -log(T * (x) -t 0 (x 0 )), we divide B into two parts

A 1 (s) = {y ∈ B | φ(s)w 2 (y, s) ≤ 1} and A 2 (s) = {y ∈ B | φ(s)w 2 (y, s) ≥ 1}. (3.12)
Accordingly, we write χ 1 (s) = χ 1 1 (s) + χ 2 1 (s), where

χ 1 1 (s) = a (p + 1)s a+1 A 1 (s) |w| p+1 log a-1 (2 + φ 2 w 2 ) log(2 + φ 2 w 2 ) - 4s p -1 ρ(y), χ 2 1 (s)
= a (p + 1)s a+1 A 2 (s) |w| p+1 log a-1 (2 + φ 2 w 2 ) log(2 + φ 2 w 2 ) - 4s p -1 ρ(y)dy.
On the one hand, by using the definition of the set A 1 (s) given in (3.12) and the expression of φ(s) in (1.21), we get, for all s ≥ -log(T * (x) -t 0 (x 0 )),

|w| p+1 log a (2 + φ 2 w 2 ) ≤ Cφ -p+1 2 (s) log |a| (2 + φ(s)) ≤ Ce -ps p-1 . (3.13)
If we integrate (3.13) over A 1 (s), we obtain

χ 1 1 (s) ≤ Ce -s . (3.14)
On the other hand, by using the definition of the φ(s) given by (1.21), we write the identity

log(2 + φ 2 w 2 ) - 4s p -1 = log(2φ -2 + w 2 ) - 2a log s p -1 . (3.15)
Furthermore, the exists S 0 > 0 such that for all s ≥ S 0 , we have φ(s) ≥ 1. Therefore, by exploiting (3.15), we write for all s ≥ max(-log(T * (x) -t 0 (x 0 )), S 0 ),

log(2 + φ 2 w 2 ) - 4s p -1 ≤ log(2 + w 2 ) + C log s. (3.16)
Also, by using the definition of the set A 2 (s) defined in (3.12), we can write for all s ≥ -log(T * (x) -t 0 (x 0 )), if y ∈ A 2 (s), we have

log(2 + φ 2 w 2 ) ≥ log(φ(s)) ≥ 2s p -1 - a log s p -1 . (3.17)
Clearly, the exists S 1 > S 0 such that for all s ≥ S 1 , we have 2s p-1 -a log s p-1 ≥ s p-1 . Therefore, by exploiting (3.16) and (3.17) we have for all s ≥ max(-log(T * (x) -t 0 (x 0 )), S 1 ),

χ 2 1 (s) ≤ C s a+2 B |w| p+1 log a (2 + φ 2 w 2 ) log(2 + w 2 )ρ(y)dy + C log s s a+2 B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy. (3.18)
Adding (3.1) and (3.18) we have for all s ≥ max(-log(T * (x) -t 0 (x 0 )), S 1 ),

χ 2 1 (s) ≤ K 15 s a+ 7 4 B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + K 15 s 7 4 . (3.19)
Note that, by using the fact χ Finally, by using the following basic inequality

1 (s) = χ 1 1 (s) + χ 2 1 (s), (3.14) and (3.19)
ab ≤ νa 2 + 1 ν b 2 , ∀ν > 0, (3.23)
and the expression of γ(s) defined in (1.20), we write, for all s ≥ -log T * (x) -t 0 (x 0 )

χ 3 (s) + χ 4 (s) ≤ α 2 B (∂ s w) 2 ρ(y) 1 -|y| 2 dy + C s 2 B |∇w| 2 (1 -|y| 2 ) + w 2 ρ(y)dy. (3.24)
The result (3.8) derives immediately from (3.9), (3.24), (3.20), (3.22), and the identity (3.9), which ends the proof of Lemma 3.2

Let us now recall the following result from [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF], where we write an estimate on the functional J(w(s), s) defined by: J(w(s), s) = - Proof: See Lemma 2.2 in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF].

With Lemmas 3.2 and 3.3, we are in a position to state and prove Theorem 1', which is a uniform version of Theorem 1 for x near x 0 .

Theorem 1' (Existence of a Lyapunov functional for equation (1.18)) Consider u a solution of (1.1) with blow-up graph Γ : {x → T (x)} and x 0 a non characteristic point. Then there exists t 1 (x 0 ) ∈ [0, T (x 0 )) such that, for all T 0 ∈ (t 1 (x 0 ), T (x 0 )], for all s ≥ -log(T 0 -t 1 (x 0 )) and x ∈ R, where |x -x 0 | ≤ T -t δ 0 (x 0 ) , we have L(w(s + 1), s + 1) -L(w(s), s) ≤ -α

s+1 s B (∂ s w) 2 ρ(y) 1 -|y| 2 dydτ, (3.28)
where w = w x,T * (x) and T * (x) is defined in (2.4). .

Proof of

Again, choosing S 2 > -log(T (x 0 ) -t 0 (x 0 )) large enough, this implies that for all s ≥ max(-log(T * (x) -t 0 (x 0 )), S 2 ), we have A simple integration between s and s + 1 ensures the result (3.28), where t 1 (x 0 ) = max(T (x 0 ) -e -S 2 , t 0 (x 0 )). (3.33) This concludes the proof of Theorem 1'.

d ds L 0 (w(s), s) ≤ -α B (∂ s w) 2 ρ(
We now claim the following lemma:

Lemma 3.4. There exists S 3 ≥ S 2 such that, if L(w(s 3 ), s 3 ) < 0 for some s 3 ≥ max(S 3 , -log(T * (x) -t 1 (x 0 ))), then w blows up in some finite time s 4 > s 3 .

Proof: The argument is the same as in Lemma 3.4 in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF].

Proof of Theorem 2

In this subsection, we prove Theorem 2. Note that the lower bound follows from the finite speed of propagation and the wellposedness in H 1 × L 2 . For a detailed argument in the similar case of equation (1.8) in [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] (see Lemma 3.1, page 1136). We consider u a solution of (1.1) with blow-up graph Γ : {x → T (x)} and x 0 is a non-characteristic point. Let

t 2 (x 0 ) = max(T (x 0 ) -e -S 3 , t 1 (x 0 )). (3.34) Given some T 0 ∈ (t 2 (x 0 ), T (x 0 )], for all x ∈ R is such that |x -x 0 | ≤ T 0 -t 2 (x 0 ) δ 0 (x 0 )
, where δ(x 0 ) is defined in (1.7), we aim at bounding (w, ∂ s w)(s) H 1 ×L 2 (B) for s large.

As in [START_REF]A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations[END_REF][START_REF] Hamza | The blow-up rate for strongly perturbed semilinear wave equations[END_REF], by combining Theorem 1' and Lemma 3.4 we get the following bounds: Corollary 3.5. (Bound on L 0 (w(s), s)). For all T 0 ∈ (t 2 (x ), T (x 0 )], for all s ≥ -log(T 0 -t 2 (x 0 )) and x ∈ R N where |x -x 0 | ≤ T 0 -t δ 0 (x 0 ) , we have

-C ≤ L 0 (w(s), s) ≤ CL 0 (w(s 2 ), s2 ) + C, (3.35)
where s2 = -log(T * (x) -t 2 (x 0 )).

Moreover, for all s ≥ -log(T * (x) -t 2 (x 0 )), we have

s+1 s B (∂ s w) 2 ρ(y) 1 -|y| 2 dyds ≤ K 16 , (3.36)
where K 16 = K 16 (a, p, T * (x), (u(t 2 ), u t (t 2 )) H 1 ×L 2 (B(x 0 , T 0 -t 2 (x 0 ) δ 0 (x 0 ) )) ), C = C(a, p) and δ 0 (x 0 ) ∈ (0, 1) is defined in (1.7). Remark 3.2. Using the definition of (1.17) of w x,T * (x) = w, we write easily

L 0 (w( s 2 ), s 2 ) ≤ K 17 , (3.37)
where

K 17 = K 17 (T (x 0 ) -t 2 (x 0 ), (u(t 2 (x 0 )), ∂ t u(t 2 (x 0 ))) H 1 ×L 2 (B(x 0 , T (x 0 )-t 2 (x 0 ) δ 0 (x 0 )
)) ).

Starting from these bounds, the proof of Theorem 2 is similar to the proof in [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF] (see Corollary 2.3, page 1151) and [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] (see Proposition 2.1, page 1134) except for the treatment of the nonlinear terms and of the perturbation terms. In our opinion, handling these terms is straightforward in all the steps of the proof, except for the first step, where we bound the time averages of the nonlinear term and second step, where we remove the time averages. However, the third step where we conclude the boundedness of the H 1 loc,u (R N ) norm of solution of equation (1.18) from Proposition 3.6 is the same as in Proposition 2.1 (up to some very minor changes). For that reason, we only give the first two step and refer to the proof in [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF] (see Corollary 2.3, page 1151) and [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] (see Proposition 2.1, page 1134) and the similar part in section 2 in this paper for the remaining steps in the proof of Theorem 2. This is the step we prove here. Proposition 3.6. For all s ≥ 1 -log(T * (x) -t 3 (x 0 )), for some t

3 (x 0 ) ∈ [t 2 (x 0 ), T (x 0 )), 1 s a s+1 s B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dydτ ≤ K 18 . (3.38)
Proof: For s ≥ 1 -log(T * (x) -t 2 (x 0 )), let us work with time integrals between s 1 et s 2 where s 1 ∈ [s -1, s] and s 2 ∈ [s + 1, s + 2]. By integrating the expression (1.24) of L 0 (w(s), s) in time between s 1 and s 2 , where s 2 > s 1 > -log(T * (x) -t 2 (x 0 )), we obtain: Note that, by using the identity (3.11), we get .

s 2 s 1 L 0 (w(s), s)ds = s 2 s 1 B 1 2 (∂ s w) 2 + p + 1 (p -1) 2 w 2 -e -2(p+1)s p-1 s 2a p-1 F (φw) ρ(y)dyds + 1 2 s 2 s 1 B (|∇w| 2 -
e -2(p+1)s p-1 s 2a p-1 φw 2 f (φw) -F (φw) = p -1 2 e -2(p+1)s p-1 s 2a p-1 F (φw) (3.41) - p + 1 2 e -2(p+1)s p-1 s 2a p-1 F 1 (φw) + F 2 (φw
We claim that Proposition 3.6 follows from the following Lemma where we control the space-time integral of the nonlinear term of w and all the terms on the right-hand side of the relation (3.42) in terms of the left-hand side:

Lemma 3.7. For all s ≥ 1 -log(T * (x) -t 3 (x 0 )), for some t 3 (x 0 ) ∈ [t 2 (x 0 ), T (x 0 )), for all ν 0 > 0, for all ε ∈ (0, 1), e -2(p+1)s p-1 s 2a p-1 F (φw)ρ(y)dyds. Now, we can use the fact that s 1 ≥ -1 -log(T * (x) -t 3 (x 0 )) ≥ -1 -log(T (x 0 ) -t 3 (x 0 )) and we choose T (x 0 ) -t 3 (x 0 ) small enough, so that C s 1 ≤ 1 -1-log(T (x 0 )-t 3 (x 0 )) ≤ 1 4 . If we choose ν 0 small enough so that K 19 ν 0 ≤ 1 4 , we obtain Since [s, s + 1] ⊂ [s 1 , s 2 ], we derive from (A.4) that (3.38).

It remains to prove Lemma 3.7.

Proof of Lemma 3.7: By (A.9) and (A.10), we can write easily (3.43) and (3.44). Thanks to (3.43), we can adapt with no difficulty the proof in the unperturbed case [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF][START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] (up to some very minor changes), in order to get the proof of the estimates (see the appendix in [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF] for a proof), we easily conclude (3.50) and (3.51).

Finally, it remains only to control the terms A 5 and A 6 . Note from (A.4), (A.5) and (A. The result (3.52) follows immediately from (3.53). This concludes the proof of Lemma 3.7 and Proposition 3.6 too. Proof of Theorem 2 : Since the derivation of the boundedness of the H 1 loc,u (R N ) norm of solution of equation (1.18) from Proposition 3.6 is the same as in Proposition 2.1 (from the estimates (2.1), (2.2) (2.3) and (2.5) (up to some very minor changes). Moreover, thanks to the estimate (3.35), the boundedness of the H 1 loc,u (R N ) norm, we prove easily the boundedness of L 2 loc,u (R N ) norm of ∂ s w with the ball B(0, 1 2 ). Thanks to the covering technique (we refer the reader to Merle and Zaag [START_REF]Blow-up rate near the blow-up surface for semilinear wave equations, Internat[END_REF] (pure power case) and Hamza and Zaag in Lemma 2.8 in [START_REF] Hamza | A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case[END_REF]), we easily extend this estimate from B(0, 1 2 ) to B. This concludes the proof of Theorem 2.

A Some elementary lemmas.

Let f , F , F 2 be the functions defined in (1.2), (1.25) and (3.11). Clearly, we have

  w(y, τ )) 2 dydτ. (2.14) Thanks to (2.14), the classical inequality x p+3-ε 2

2 .

 2 By using(2.16) and the Sobolev embeddingH 1 (B) → L r (B) we get B |w(y, s)| p+1+ε dy ≤

1

 1 (φw)| + |F 2 (φw)| ≤ C + C φw s 2 f (φw). (3.21) By (3.9), (3.21) and (2.10), we have, for all s ≥ -log(T * (x) -t 0 (x 0 )), χ 2 (s) ≤ C s a+2 B |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + Ce -2s . (3.22)

  4s

  (3.30) and(3.31) lead to the following crucial estimate: d ds L(w(s), s) ≤ -α exp p + 3 √ s B (∂ s w) 2 ρ(y) 1 -|y| 2 dy + K 15 exp Since we have 1 ≤ exp p+3 √ s ≤ exp p+3 √ S 2 , we then choose θ large enough, so that K 15 exp p+3 √ s -3θ 4 ≤ 0, which yields, for all s ≥ max(-log(T * (x) -t 0 (x 0 )), S 2 ), d ds L(w(s), s) ≤ -α B (∂ s w) 2 ρ(y) 1 -|y| 2 dy.

1 F

 1 (φw)ρ(y)dyds ≤ K 19 .

( 3 .

 3 [START_REF] Sasaki | Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity[END_REF],(3.46),(3.47),(3.48) and(3.49). Also, by using (3.43) and the Hardy inequalityB w 2 |y| 2 ρ(y) 1 -|y| 2 dy ≤ C B |∇w| 2 (1 -|y| 2 )ρ(y)dy + C B w 2 ρ(y)dy.

  6) that|F 1 (φw)| + |F 2 (φw)| ≤ C + C F (φw) s . (3.53)

  From (2.5), the definition (2.6) of H m 0 (w(s), s), we see that for all s ≥ -log(T * (x)t 0 (x 0 )),

	where w = w x,T B |∇w(y, s)| 2 (1 -|y| 2 )ρ(y)dy +	B	(∂ s w(y, s)) 2 wdy -	2m 0 s B	w∂ s wρ(y)dy
				≤ 2	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dy + 2K 1 s q .	(2.21)
						B
	By the use of the basic inequality 2ab ≤ a 2 + b 2 , we write
	2m 0 s B	w∂ s wρ(y)dy ≤	B	(∂ s w(y, s)) 2 wdy +	(m 0 ) 2 s 2	B	w 2 ρ(y)dy.	(2.22)
	Plugging (2.22) and (2.3) into (2.21), we obtain
									|∇w| 2 dy,	(2.20)
									1 2 )

* (x) (y, s) is defined in (1.17), with T * (x) given in (2.4) and s = -log(T * (x) -t).

B |∇w(y, s)| 2 (1 -|y| 2 )ρ(y)dy ≤ 2 B e -2(p+1)s p-1 s 2a p-1 F (φw)ρ(y)dy + K 6 s q .

  |w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + Ce -2s .

	Lemma 3.3. For all s ≥ max(-log T * (x), 1), we have
	d ds	J(w(s), s) ≤	p + 3 2s	E(w(s), s) -	p + 7 4s B	(∂ s w) 2 ρ(y)dy	(3.26)
								-	p -1 4s B	(|∇w| 2 -(y.∇w) 2 )ρ(y)dy -	p + 1 2(p -1)s B	w 2 ρ(y)dy
								-	p -1 2(p + 1)s a+1	B	|w| p+1 log a (2 + φ 2 w 2 )ρ(y)dy + Σ 2 (s),
	where Σ 2 (s) satisfies
	Σ 2 (s) ≤	C √ s B	(∂ s w) 2 ρ(y) 1 -|y| 2 dy +	s	C √	s B	|∇w| 2 (1 -|y| 2 )ρ(y)dy	(3.27)
			+	s	C √	s B	w 2 ρ(y)dy +	C s a+2	B
									1 s B	w∂ s wρ(y)dy.	(3.25)

  |y.∇w| 2 )ρ(y)dyds -By multiplying the equation (1.18) by wρ(y) and integrating both in time and in space over B × [s 1 , s 2 ] we obtain the following identity, after some integration by parts:

								s 2 s 1	s	1 √	s B w∂ s wρ(y)dyds.
								(3.39)
		B	w∂ s w + (	p + 3 2(p -1)	-N )w 2 ρ(y)dy	s 2 s 1	=	s 2 s 1 B (∂ s w) 2 ρ(y)dyds (3.40)
	-		s 2 s 1 B	|∇w| 2 -(y.∇w) 2 ρ(y)dyds -	2p + 2 (p -1) 2	s 2 s 1 B w 2 ρ(y)dyds
	+		s 2 s 1 B e -2ps p-1 s		s 2 s 1 B w∂ s w	|y| 2 ρ(y) 1 -|y| 2 dyds
	+2	s 2 s 1 B y.∇w∂ s wρ(y)dyds +	2a p -1	s 2 s 1	B	1 s	y.∇wwρ(y)dyds
	+	s 2 s 1 B γ(s)w 2 ρ(y)dyds +	2a p -1	s 2 s 1	B	1 s	∂ s wwρ(y)dyds.

a p-1 wf (φw)ρ(y)dyds -2α

  ) .

		-	1 2	s 2 s 1 B γ(s)w 2 ρ(y)dyds	-	a p -1	s 2 s 1	B	1 s	∂ s wwρ(y)dyds	(3.42)
							A 2			A 3
		+		s 2 s 1	s	1 √	s B w∂ s wρ(y)dyds	+	p + 1 2	s 1 B s 2	e -2(p+1)s p-1 s	2a p-1 F 1 (φw)ρ(y)dyds
								A 4		A 5
		+	p + 1 2	s 1 B s 2	e -2(p+1)s p-1 s	2a p-1 F 2 (φw)ρ(y)dyds
										A 6
	By combining the identities (3.39), (3.40) and exploiting (3.41), we obtain
		p -1 2		s 1 B s 2	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds
	=	1 2 B	w∂ s w + (	p + 3 2(p -1)	-N )w 2 ρ(y)dy	s 2 s 1	-	s 2 s 1 B (∂ s w) 2 ρ(y)dyds
		+		s 2 s 1 L 0 (w(s), s)ds + α	s 2 s 1 B w∂ s w	|y| 2 ρ(y) 1 -|y| 2 dyds
		-	s 2 s 1 B y.∇w∂ s wρ(y)dyds -	a p -1	s 2 s 1	B	1 s	y.∇wwρ(y)dyds
											A 1

  w(y, s 1 )) 2 + (∂ s w(y, s 2 )) 2 ρ(y)dy ≤ K 19 , (3.49)

	Indeed, from (3.42) and this Lemma, we deduce that
	s 2 s 1	B	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds ≤	K 19 ν 0	+ (K 19 ν 0 +	C s 1	)	s 2 s 1	B
		s 2 s 1 B	|y.∇w∂ s w|ρ(y)dyds ≤	K 19 ν 0	+ K 19 ν 0	s 1 s 2	B	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds,
	sup s∈[s 1 ,s 2 ] B w 2 (y, s)ρ(y)dy ≤	K 19 ν 0	+ K 19 ν 0	s 1 s 2	B	e -2(p+1)s p-1 s	(3.45) p-1 F (φw)ρ(y)dyds, (3.46) 2a
		s 2 s 1 B w∂ s w	|y| 2 ρ(y) 1 -|y| 2 dyds ≤		K 19 ν 0	+ K 19 ν 0	s 1 s 2	B	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds,
											(3.47)
											K 19
											ν 0
											+ K 19 ν 0	s 2	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds,	(3.48)
											s 1	B
				|A 1 | ≤	K 19 ν 0	+ (K 19 ν 0 +	C s 1	)	s 1 s 2	B	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds,	(3.50)
		|A 2 | + |A 3 | + |A 4 | ≤	K 19 ν 0	+ K 19 ν 0	s 1 s 2	B	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds,	(3.51)
					|A 5 | + |A 6 | ≤ C +	C s 1	s 1 s 2	B	e -2(p+1)s p-1 s	2a p-1 F (φw)ρ(y)dyds.	(3.52)

B |w| p+1-ε ρ(y)dy ≤ K 19 + C B e -2(p+1)s p-1 s 2a p-1 F (φw)ρ(y)dy,

(3.43)

B e -2(p+1)s p-1 s 2a p-1 F (φw)ρ(y)dy ≤ K 19 + C B |w| p+1+ε ρ(y)dy,

(3.44)

B |w∂ s w|ρ(y)dy ≤ B (∂ s w) 2 ρ(y)dy + B (∂ s

Lemma A.1. Let q > 1, u 0 |v| q-1 v log a (2 + v 2 )dv ∼ |u| q+1 q + 1 log a (2 + u 2 ), as |u| → ∞, (A.1)

Proof. See Lemma A.1 in [START_REF]The blow-up rate for a non-scaling invariant semilinear wave equations[END_REF].

Thanks to (A.1), (A.2) and (A.3), we can state and prove the following estimates: Proof. Note that (A.4) obviously follows from (A.2). In order to derive estimates (A.5) and (A.6), considering the first case z 2 φ(s) ≥ 4, then the case z 2 φ(s) ≤ 4, we would obtain (A.5) and (A.6) by using (A.1), (A.2) and(A.3). Similarly, by taking into account the inequality log a (2 + u 2 ) ≤ C(ε) + |u| ε , we conclude easily (A.7), (A.8), (A.9) and (A.10). This ends the proof of Lemma A.2.