Blow-up results for semilinear wave equations in the super-conformal case

We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Vi\c{s}an [6]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [5] for perturbations of the pure power case with lower order terms.

We would like to mention that equation (1.1) encompasses the case of the following nonlinear Klein-Gordon equation In order to keep our analysis clear, we only give the proof for the following non perturbed equation and refer the reader to [4] and [5] for straightforward adaptations to equation (1.1).
The Cauchy problem of equation (1.3) is solved in H 1 loc × L 2 loc . This follows from the finite speed of propagation and the the wellposedness in H 1 ×L 2 , valid whenever 1 < p < p S . The existence of blow-up solutions for the associated ordinary differential equation of (1.3) is a classical result. By using the finite speed of propagation, we conclude that there exists a blow-up solution u(t) of (1.3) which depends non trivially on the space variable. In this paper, we consider a blow-up solution u(t) of (1.3), we define (see for example Alinhac [1] and [2]) Γ as the graph of a function x → T (x) such that the domain of definition of u is given by The set D u is called the maximal influence domain of u. Moreover, from the finite speed of propagation, T is a 1-Lipschitz function. The graph Γ is called the blow-up graph of u. Let us first introduce the following non-degeneracy condition for Γ. If we introduce for all x ∈ R N , t ≤ T (x) and δ > 0, the cone then our non degeneracy condition is the following: x 0 is a non characteristic point if We aim at studying the growth estimate of u(t) near the space-time blow-up graph in the super-conformal case (where p c < p < p S ). Let us briefly mention some results concerning the blow-up rate of solutions of semilinear wave equations. The first result valid for general solutions is due to Merle and Zaag in [8] (see also [7] and [9]) who proved, that if 1 < p ≤ p c and u is a solution of (1.3), then the growth estimate near the space-time blow-up graph is given by the associated ODE. In [4] and [5], we extend the result of Merle and Zaag to perturbed equations of type (1.1) under some reasonable growth estimates on f and g in (1.1) (see hypothesis (H f ) and (H g )). Note that, in all these papers, the method crucially relies on the existence of a Lyapunov functional in similarity variables established by Antonini and Merle [3]. Recently, Killip, Stovall and Vişan in [6] have shown, among other results, that the results of Merle and Zaag remain valid for the semilinear Klein-Gordon equation (1.2). Moreover, they consider also the case where p c < p < p S and prove that, if u is a solution of (1.2), then for all and for all t ∈ (0, T (x 0 )], Moreover, if x 0 is a non characteristic point, then they use a covering argument to obtain the same estimates with the ball B(x 0 , T (x 0 )−τ 2 ) replaced by the ball B(x 0 , T (x 0 ) − τ ) in the inequalities (1.6) and (1.7).
Here, we obtain a better result thanks to a different method based on the use of self-similar variables. This method allows us to improve the results of [6] as we state in the following: THEOREM 1 (Growth estimate near the blow-up surface for Eq. (1.1)). If u is a solution of (1.1) with blow-up graph Γ : {x → T (x)}, then for all x 0 ∈ R N and t ∈ [0, T (x 0 )), we have Moreover, for all t ∈ (0, T (x 0 )], we have and If in addition x 0 is a non characteristic point, then we have for all t ∈ (0, T (x 0 )], Moreover, we have , the finite speed of propagation and the fact that x 0 is a non characteristic point): there exist ε 0 > 0, such that ii) In Theorem 1, we improve recent results of Killip, Stovall and Vişan in [6]. More precisely, we obtain a better estimate in (1.8) and if x 0 is non characteristic point we have the better estimate (1.11). iii) Up to a time dependent factor, the expression in (1.12) is equal to the main terms of the energy in similarity variables (see (1.20)). However, even with this improvement, we think that our estimates are still not optimal. iv) The constant K 1 , and the rate of convergence to 0 of the different quantities in the previous theorem and in the whole paper, depend only on N, p and the upper bound on T (x 0 ), 1/T (x 0 ), and the initial data Our method relies on the estimates in similarity variables introduced in [3] and used in [7], [8] and [9]. More precisely, given (x 0 , T 0 ) such that 0 < T 0 ≤ T (x 0 ), we introduce the following self-similar change of variables: This change of variables transforms the backward light cone with vortex (x 0 , T 0 ) into the infinite cylinder (y, s) ∈ B × [− log T 0 , +∞). 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 → +∞.
From (1.3), the function w x 0 ,T 0 (we write w for simplicity) satisfies the following equation for all y ∈ B ≡ B(0, 1) and s ≥ − log T 0 : Putting this equation in the following form the key idea of our paper is to view this equation as a perturbation of the conformal case (corresponding to η = 0) already treated in [5] with the term 2ηy · ∇w. Of course, this term is not a lower order term with respect to the nonlinearity. For that reason, we will have exponential growth rates in the w setting. Let us emphasize the fact that our analysis is not just a trivial adpatation of our previous work [5].
The equation (1.14) will be studied in the Hilbert space H In the conformal case where p = p c , Merle and Zaag [8] proved that is a Lyapunov functional for equation (1.14). When p > p c , we introduce where and η is defined in (1.16). Finally, we define the energy function as (1.20) The proof of Theorem 1 crucially relies on the fact that F (w, s) is a Lyapunov functional for equation (1.14) on the one hand, and on the other hand, on a blowup criterion involving F (w, s). Indeed, with the functional F (w, s) and some more work, we are able to adapt the analysis performed in [8]. In the following, we show that F (w, s) is a Lyapunov functional: PROPOSITION 1.2 (Existence of a decreasing functional for Eq. (1.14)).
This paper is organized as follows: In section 2, we prove Proposition 1.2. Using this result, we prove Theorem 1 in section 3.
2 Existence of a decreasing functional for equation (1.14) and a blow-up criterion Consider u a solution of (1.3) with blow-up graph Γ : {x → T (x)}, and consider its self-similar transformation w x 0 ,T 0 defined at some scaling point ( . This section is devoted to the proof of Proposition 1.2. We proceed in two parts: • In subsection 2.1, we show the existence of a decreasing functional for equation (1.14).
• In subsection 2.2, we prove a blow-up criterion involving this functional.

Existence of a decreasing functional for equation (1.14)
In this subsection, we prove that the functional F (w, s) defined in (1.20) is decreasing. More precisely we prove that the functional F (w, s) satisfies the inequality (1.21). Now we state two lemmas which are crucial for the proof. We begin with bounding the time derivative of E 0 (w) defined in (1.17) in the following lemma. Since we see from integration by parts that this concludes the proof of Lemma 2.1.
We are now going to prove the following estimate for the functional I(w): Proof: Note that I(w) is a differentiable function for all s ≥ − log T 0 and that d ds By using equation (1.15) and integrating by parts, we have Then by integrating by parts, we have

A blow-up criterion
We finish the proof of Proposition 1.2 here. More precisely, for all x 0 ∈ R N and T 0 ∈ (0, T (x 0 )], we prove that We give the proof only in the case where x 0 is a non characteristic point. Note that the case where x 0 is a characteristic point can be done exactly as in Appendix A page 119 in [10].
• (B) By construction, w δ is also a solution of equation (1.14).
Proof: The first three estimates are a direct consequence of Proposition 1.2. As for the last estimate, by introducing f (y, s) = e −ηs w(y, s), we see that the dispersion estimate (3.9) can be written as follows: In particular, we have . Now, we control all the terms on the right-hand side of the relation (3.13): Note that the first term is negative, while the second term is bounded because of the bound (3.7) on the energy F (w, s). Since |y.∇w| ≤ |y||∇w|, we can say that the third is also negative. Remark that (3.8) implies that the fourth term is also bounded. Finally, it remains only to control the term A(s).
Combining the Cauchy-Schwarz inequality, the inequality ab ≤ εa 2 + 1 4ε b 2 , and the fact that N ≥ 2 and η ∈ [0, 1], we write (3.14) Now, we are able to conclude the proof of the inequality (3.1). For this, we combine (3.13), (3.14) and the above-mentioned arguments for the first four terms to get The desired bound in (3.1) follows then from (3.15).
Since the derivation of Theorem 1 from Proposition 3.1 is the same as in [8] (up to some very minor changes), this concludes the proof of Theorem 1.