Solutions with prescribed local blow-up surface for the nonlinear wave equation

TL;DR

This paper demonstrates that any sufficiently smooth space-like hypersurface in spacetime can be locally realized as the blow-up surface of a finite-energy solution to the focusing nonlinear wave equation, extending previous construction methods.

Contribution

It introduces a method to prescribe local blow-up surfaces for solutions of the nonlinear wave equation, generalizing prior work to include arbitrary space-like hypersurfaces.

Findings

Any smooth space-like hypersurface can be locally realized as a blow-up surface.

Construction of solutions with prescribed blow-up surfaces is possible for dimensions 1 to 4.

Solutions with finite energy can be obtained using the developed approach.

Abstract

We prove that any sufficiently differentiable space-like hypersurface of ${\mathbb R}^{1+N} $ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation $\partial_{tt} u - \Delta u=|u|^{p-1} u$ on ${\mathbb R} \times {\mathbb R} ^N$, for any $1\leq N\leq 4$ and $1 < p \le \frac {N+2} {N-2}$. We follow the strategy developed in our previous work [arXiv 1812.03949] on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blowup on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at $t=0$ for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at $t=0$. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with $H^2\times H^1$ solutions for the transformed problem.

Equations

$${\partial }_{tt}u-\Delta u=|u{|}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$

$$q_0=2\lfloor \frac{2p+2}{p-1}\rfloor +3.$$

$$\varphi(0)=0\quad\mbox{and}\quad|\nabla\varphi(x)|<1\mbox{ for all $x\in{\mathbb{R}}^{N}$.}$$

$$\underset{t\uparrow {\tau }_0+\phi (x_0)}{\mathrm{liminf}}\frac{1}{{\tau }_0+\phi (x_0)-t}{\int }_t^{{\tau }_0+\phi (x_0)}d{t}^{\prime }{\int }_{\{|x-x_0|<\sigma ({\tau }_0+\phi (x_0)-t^{\prime })\}}|{\partial }_{t}u{|}^{2}dx>0.$$

$${\mathbb{R}}_{+}^{1+N}=[0,+\infty )\times {\mathbb{R}}^N.$$

$$C(t,x)=\left\{(s,y)\in {\mathbb{R}}_{+}^{1+N}\text{mbox}suchthat|x-y|<t-s\right\}.$$

$$\varphi (x)=\sup \left\{t\ge 0\text{mbox}suchthat(t,x)\in \Omega \right\}.$$

$$E(x_{0},R,\tau)=\left\{(t,x)\in{\mathbb{R}}_{+}^{1+N}\mbox{ such that $0\leq t<\tau$ and $|x-x_{0}|<R-t$}\right\}.$$

$${\tilde{u}}_0=u_0\text{mbox}and{\tilde{u}}_1=u_1\text{on B(x0,R)}.$$

$$x\in {\mathbb{R}}^N\mapsto \varphi (x)=\sup \left\{t\ge 0\text{mbox}suchthat(t,x)\in {\Omega }_{\max }(u_0,u_1)\right\}\in (0,\infty ).$$

$$(1-|\nabla \psi {|}^2){\partial }_{ss}v-2\nabla \psi \cdot \nabla {\partial }_{s}v-(\Delta \psi ){\partial }_{s}v-\Delta v=f(v)$$

$$\chi \equiv 1$$

$$\displaystyle\Big{|}F(u+v)-F(v)-F^{\prime}(u)v-\frac{1}{2}F^{\prime\prime}(u)v^{2}\Big{|}$$

$$|(f(u+v)-f(u)-f^{\prime }(u)v)v|$$

$$|f^{\prime }(u+v)-f^{\prime }(u)|$$

$$\Big{|}f(u+v)-f(u)-f^{\prime}(u)v-\frac{1}{2}f^{\prime\prime}(u)v^{2}\Big{|}\lesssim u^{-1}|v|^{p+1}+u^{p-\bar{p}-1}|v|^{\bar{p}+1}.$$

$$|\beta |=\sum _{j=1}^N{\beta }_j,\beta !=\prod _{j=1}^N({\beta }_j!),$$

$$\displaystyle x^{\beta}=\prod_{j=1}^{N}x_{j}^{\beta_{j}},\quad\partial_{x}^{\beta}=\frac{\partial^{|\beta|}}{\partial_{x_{1}}^{\beta_{1}}\ldots\partial_{x_{N}}^{\beta_{N}}}\quad\mbox{for $|\beta|>0$,}\quad\partial_{x}^{0}=\mathop{\textnormal{Id}}.$$

$$\left(\genfrac{}{}{0ex}{}{\beta }{{\beta }^{\prime }}\right)=\prod _{j=1}^N\left(\genfrac{}{}{0ex}{}{{\beta }_j}{{\beta }_j^{\prime }}\right)=\frac{\beta !}{({\beta }^{\prime }!)(\beta -{\beta }^{\prime })!}.$$

$${\partial }_x^{\beta }\left(gh\right)=\sum _{{\beta }^{\prime }\le \beta }\left(\genfrac{}{}{0ex}{}{\beta }{{\beta }^{\prime }}\right)\left({\partial }^{{\beta }^{\prime }}g\right)\left({\partial }^{\beta -{\beta }^{\prime }}h\right).$$

$${\partial }_x^{\beta }(f\circ g)=\sum _{r=1}^n\left(f^{(r)}\circ g\right)\sum _{P(\beta ,r)}(\beta !)\prod _{\ell =1}^n\frac{{\left({\partial }_x^{{\beta }_{\ell }}g\right)}^{r_{\ell }}}{(r_{\ell }!)({\beta }_{\ell }!{)}^{r_{\ell }}}$$

$$P(\beta ,r)$$

$$\displaystyle\qquad r_{\ell}=0\mbox{ and }\beta_{\ell}=0\mbox{ for $1\leq\ell\leq n-m$};\,r_{\ell}>0\mbox{ for $n-m+1\leq\ell\leq n$};$$

$$\displaystyle\qquad\mbox{and }0\prec\beta_{n-m+1}\prec\cdots\prec\beta_{n}\mbox{ are such that }\sum_{\ell=1}^{n}r_{\ell}=r,\ \sum_{\ell=1}^{n}r_{\ell}\beta_{\ell}=\beta\Big{\}}.$$

$$|\mathbf{\nu }|=\alpha +|\beta |,\mathbf{\nu }!=\alpha !\beta !,{\partial }^{\mathbf{\nu }}={\partial }_s^{\alpha }{\partial }_x^{\beta }.$$

$$\left(\genfrac{}{}{0ex}{}{\mathbf{\nu }}{{\mathbf{\nu }}^{\prime }}\right)=\left(\genfrac{}{}{0ex}{}{\alpha }{{\alpha }^{\prime }}\right)\left(\genfrac{}{}{0ex}{}{\beta }{{\beta }^{\prime }}\right).$$

$${\partial }^{\mathbf{\nu }}\left(gh\right)=\sum _{{\mathbf{\nu }}^{\prime }\le \mathbf{\nu }}\left(\genfrac{}{}{0ex}{}{\mathbf{\nu }}{{\mathbf{\nu }}^{\prime }}\right)\left({\partial }^{{\mathbf{\nu }}^{\prime }}g\right)\left({\partial }^{\mathbf{\nu }-{\mathbf{\nu }}^{\prime }}h\right).$$

$${\partial }^{\mathbf{\nu }}(f\circ g)=\sum _{r=1}^n\left(f^{(r)}\circ g\right)\sum _{P(\mathbf{\nu },r)}(\mathbf{\nu }!)\prod _{\ell =1}^n\frac{{\left({\partial }^{{\mathbf{\nu }}_{\ell }}g\right)}^{r_{\ell }}}{(r_{\ell }!)({\mathbf{\nu }}_{\ell }!{)}^{r_{\ell }}}$$

$$P(\mathbf{\nu },r)$$

$$\displaystyle\qquad r_{\ell}=0\mbox{ and }{\boldsymbol{\nu}}_{\ell}=0\mbox{ for $1\leq\ell\leq n-m$};\,r_{\ell}>0\mbox{ for $n-m+1\leq\ell\leq n$};$$

$$\displaystyle\qquad\mbox{and }0\prec{\boldsymbol{\nu}}_{n-m+1}\prec\cdots\prec{\boldsymbol{\nu}}_{n}\mbox{ are such that }\sum_{\ell=1}^{n}r_{\ell}=r,\ \sum_{\ell=1}^{n}r_{\ell}{\boldsymbol{\nu}}_{\ell}={\boldsymbol{\nu}}\Big{\}}.$$

$$\psi (x)=0$$

$$u(t,x)=v(s,x),s=\psi (x)-t$$

$${\partial }_{tt}u$$

Full text

L. Zhao was partially supported by the NSFC Grant of China No. 11771415

Dedicated to Laurent Véron on the occasion of his 70th birthday

1. 1 Introduction

2. 1.1 Main result

3. 1.2 Definition of the maximal influence domain

4. 1.3 Previous results

5. 1.4 Strategy of the proof of Theorem 1.1

6. 1.5 Notation

7. 2 Blow up ansatz

8. 2.1 Change of variables

9. 2.2 First blow up ansatz

10. 2.3 Refined blow up ansatz

11. 3 Construction of a solution of the transformed equation $\eqref{eq:v}$

12. 4 Proof of Theorem 1.1

13. 4.1 Cut-off of the local hypersurface

14. 4.2 Construction of the function $\psi$

15. 4.3 Definition of an appropriate solution of the transformed equation

16. 4.4 Returning to the original variable

17. 4.5 Choice of a solution of the nonlinear wave equation

18. 4.6 Blowup on the local hypersurface and end of the proof

19. A The wave equation (2.2)

20. A.1 The associated semigroup

21. A.2 The nonlinear equation

22. A.3 Regularity

23. A.4 The case of equation (3.7)

24. B Uniqueness on light cones

1. Introduction

1.1. Main result

We consider the nonlinear energy-subcritical or -critical wave equation

[TABLE]

for $N\geq 1$ and $1<p\leq\frac{N+2}{N-2}$ ($1<p<\infty$ if $N=1,2$). For simplicity, we restrict ourselves to space dimensions $1\leq N\leq 4$. In this case, it is well-known that the Cauchy problem for (1.1) is locally well-posed in the energy space $H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$. (See Remark B.1.)

When a solution $u$ with initial data at $t=t_{0}$ is not globally defined ([14, 23, 1]), we introduce its maximal influence domain whose upper boundary is a $1$-Lipschitz graph. See [1, Section III.2] and, for the present setting, Section 1.2.

We prove that any sufficiently differentiable space-like hypersurface of ${\mathbb{R}}^{1+N}$ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation (1.1). More precisely, our main result is the following.

Theorem 1.1**.**

Let $1\leq N\leq 4$ and $1<p\leq\frac{N+2}{N-2}$. Let

[TABLE]

Let $\varphi:{\mathbb{R}}^{N}\to{\mathbb{R}}$ be a function of class ${\mathcal{C}}^{q_{0}}$ such that

[TABLE]

There exist $\varepsilon>0$, $\tau_{0}>0$ and $(u_{0},u_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$ such that the upper boundary of the maximal influence domain of the solution $u$ of (1.1) with initial data $(u,\partial_{t}u)(0)=(u_{0},u_{1})$ contains the local hypersurface $\{(t,x):t=\tau_{0}+\varphi(x)\mbox{ and }|x|<\varepsilon\}$. Moreover, $u$ blows up on this local hypersurface in the sense that if $|x_{0}|<\varepsilon$ and $\sigma\in(|\nabla\varphi(0)|,1)$, then

[TABLE]

It follows from (1.4) that $\partial_{t}u$ concentrates on the local hypersurface $\{(t,x):t=\tau_{0}+\varphi(x)\mbox{ and }|x|<\varepsilon\}$ in the sense of $L^{2}$. In particular, this local hypersurface is a blow-up surface for the solution $u$.

Compared to previous results (see Section 1.3), Theorem 1.1 applies to any space dimension $N\leq 4$ and any subcritical or critical $p$. Moreover, our strategy is different. It mainly relies on the construction of an ansatz by elementary ODE arguments. (See Section 1.4.)

Remark 1.2**.**

In the definition of $q_{0}$ above, we use the notation $y\mapsto\lfloor y\rfloor$ for the floor function which maps $y$ to the greatest integer less than or equal to $y$. Note that $q_{0}=7$ for $p>5$ and $q_{0}\to\infty$ as $p\to 1^{+}$. See Remark 2.3 for comments on this condition.

1.2. Definition of the maximal influence domain

We adapt the presentation of [1], Chapter III (see also [24]) to the framework of $H^{1}\times L^{2}$ solutions for the energy subcritical or critical wave equation in space dimension $N\geq 1$. Let

[TABLE]

For any $(t,x)\in{\mathbb{R}}_{+}^{1+N}$, we define the open (in ${\mathbb{R}}_{+}^{1+N}$) backward cone

[TABLE]

Definition 1.3**.**

An open set $\Omega$ of ${\mathbb{R}}_{+}^{1+N}$ is called an influence domain if $(t,x)\in\Omega$ implies $\overline{C}(t,x)\subset\Omega$.

For $\Omega$ an influence domain containing $\{0\}\times{\mathbb{R}}^{N}$, define for any $x\in{\mathbb{R}}^{N}$,

[TABLE]

From the above definition, either $\phi$ is identically $\infty$, or it is finite for all $x\in{\mathbb{R}}^{N}$. In the latter case, $\phi$ is a $1$-Lipschitz continuous function.

Recall that by the Cauchy theory in the energy space $H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$, for any $(u_{0},u_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$ there exist $T>0$ and a solution $(u,\partial_{t}u)$ of (1.1) belonging to $C([0,T],H^{1}({\mathbb{R}}^{N}))\cap C([0,T],L^{2}({\mathbb{R}}^{N}))$. These solutions are unique in that class, except for the 3D critical case $p=5$, where uniqueness is known in $C([0,T],H^{1}({\mathbb{R}}^{3}))\cap C([0,T],L^{2}({\mathbb{R}}^{3}))\cap L^{8}((0,T)\times{\mathbb{R}}^{3})$. (See Remark B.1 for details.)

From the local Cauchy theory, it is standard to define the notion of maximal solution and maximal time of existence $T_{\rm max}(u_{0},u_{1})>0$; if $T_{\rm max}(u_{0},u_{1})=\infty$, the solution is globally defined, otherwise it blows up as $t\uparrow T_{\rm max}(u_{0},u_{1})$ (in a suitable norm related to the resolution of the Cauchy problem).

To define the notion of maximal influence domain corresponding to an initial data we first extend the Cauchy theory of ${\mathbb{R}}^{N}$ to truncated cones. For $x_{0}\in{\mathbb{R}}^{N}$ and $0\leq\tau\leq R$, we define

[TABLE]

Suppose that $x_{0}\in{\mathbb{R}}^{N}$ and $R>0$, and let $(u_{0},u_{1})\in H^{1}(B(x_{0},R))\times L^{2}(B(x_{0},R))$. Consider any extension $(\widetilde{u}_{0},\widetilde{u}_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$ of $(u_{0},u_{1})$ i.e. any function satisfying

[TABLE]

Next, consider the solution $(\widetilde{u},\partial_{t}\widetilde{u})$ of (1.1) corresponding to the initial data $(\widetilde{u}_{0},\widetilde{u}_{1})$ defined on a time interval $[0,\widetilde{\tau}]$ where $\widetilde{\tau}>0$, given by the above Cauchy theory. Note that if $(\check{u}_{0},\check{u}_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$ is another extension of $(u_{0},u_{1})$ and $(\check{u},\partial_{t}\check{u})$ is the corresponding solution of (1.1) on a time interval $[0,\check{\tau}]$ ($\check{\tau}>0$) then by finite speed of propagation (see Proposition B.2), the two solutions $(\widetilde{u},\partial_{t}\widetilde{u})$ and $(\check{u},\partial_{t}\check{u})$ are identically equal on the truncated cone $E(x_{0},R,\min(\widetilde{\tau},\check{\tau}))$. In this way, we have defined a notion of solution of (1.1) on $E(x_{0},R,\tau)$ for some $\tau>0$ which is independent of the extension chosen and includes a uniqueness property. From now on, for any $(u_{0},u_{1})\in H^{1}(B(x_{0},R))\times L^{2}(B(x_{0},R))$ and any $\tau>0$, we refer to the solution of (1.1) on $E(x_{0},R,\tau)$ in this sense.

By time-translation invariance of the equation and considering the map $(t,x)\in E(x_{0},R,\tau)\mapsto u(t_{0}+t,x)$, we extend this definition to any truncated cone in ${\mathbb{R}}^{1+N}_{+}$.

Now, we define the notion of solution in an influence domain.

Definition 1.4**.**

Let $(u_{0},u_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$. Let $\Omega$ be an influence domain. We say that $(u,\partial_{t}u)$ is a solution of (1.1) on $\Omega$ with initial data $(u_{0},u_{1})$ if the following hold.

1. (i)

$u\in H^{1}_{\mathrm{loc}}(\Omega)$; 2. (ii)

For any $t_{0}\geq 0$, $x_{0}\in{\mathbb{R}}^{N}$ and $R>0$ such that $[0,t_{0}]\times B(x_{0},R)\subset\Omega$, it holds $u_{|[0,t_{0}]\times B(x_{0},R)}\in C([0,t_{0}],H^{1}(B(x_{0},R)))\cap C^{1}([t_{1},t_{2}],L^{2}(B(x_{0},R)))$; moreover, $u(0)\equiv u_{0}$ and $\partial_{t}u(0)\equiv u_{1}$ on $B(x_{0},R)$; 3. (iii)

For any $(t_{0},x_{0})\in\Omega$ and $R>0$ such that $\{t_{0}\}\times B(x_{0},R)\subset\Omega$, there exists $0<\tau<R$ such that $(t,x)\in E(x_{0},R,\tau)\mapsto u(t_{0}+t,x)$ is solution of (1.1) in the above sense.

Definition 1.5**.**

For any $(u_{0},u_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$, we denote $\Omega_{\rm max}(u_{0},u_{1})$ the union of all the influence domains $\Omega$ such that there exists a solution $(u,\partial_{t}u)$ with initial data $(u_{0},u_{1})$ on $\Omega$ in the sense of Definition 1.4.

It follows that, for any initial data $(u_{0},u_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$, $\Omega_{\rm max}(u_{0},u_{1})$ is the maximal influence domain on which a (unique) solution of (1.1) with initial data $(u_{0},u_{1})$ exists. Finally, in the case $T_{\rm max}(u_{0},u_{1})<+\infty$ the upper boundary of the maximal influence domain is the graph of the $1$-Lipschitz application

[TABLE]

1.3. Previous results

Under certain assumptions, it is known that the upper boundary of the maximal influence domain is a blow-up surface in the sense that the solution blows up (at the same rate as the ODE) on the surface, and the blow-up surface is $C^{1}$. See [4, 3] and [1, Chapter III]. See also [9, 25, 26] and the references therein for further blow-up results.

Constructing solutions of the wave equation (1.1) with prescribed blow-up surface is a classical question. Results similar to Theorem 1.2 have been proved in several cases. For the wave equation with cubic nonlinearity, it is proved in [18, Theorem 10.14, p. 192] that there exist solutions (locally defined around the blow-up surface) blowing up exactly on a prescribed surface of class $H^{r}({\mathbb{R}}^{N})$ with $r>\frac{N}{2}+7$. In [22, Theorem 1.1], an analogous result is proved in space dimension $1$ for equation (1.1) for any $p>1$. For previous results, see [1, 20, 21, 17, 16].

A related question is the study of the blow-up set, which is the intersection of the blow-up surface with the hyperplane $\{t=T_{\max}\}$. In [22, Corollary 1.2], it is proved for (1.1) in space dimension $1$ that, given any compact subset $K$ of ${\mathbb{R}}$, there exist smooth initial data for which the blow-up set is precisely $K$. This result is extended in [7, Theorem 1.1] to any space dimension and any energy-subcritical $p$. See [19] for a related result.

1.4. Strategy of the proof of Theorem 1.1

We follow closely the strategy of [7] (see also [6]). It is based on the construction of an appropriate approximate solution which blows up at $t=0$, combined with an energy method for the existence of an exact solution that also blows up at $t=0$. Here, we wish to prove blowup on a local space-like hypersurface. In order to apply the previously recalled strategy, we therefore apply a change of variable to reduce the problem to blowup at $t=0$ (Section 2.1). By doing so, we are led to study the transformed equation

[TABLE]

in the dual variables $(s,y)\in{\mathbb{R}}\times{\mathbb{R}}^{N}$. The construction of an appropriate ansatz for this equation (Sections 2.2 and 2.3) is similar to the construction made in [7]. In particular, it is based on elementary ODE arguments. The energy method for this transformed equation requires a smallness condition on $\|\nabla\psi\|_{L^{\infty}}$, and yields an existence time that depends on $\psi$. See Section 3. This smallness condition can be met through a localization argument (Section 4.1) and a Lorentz transform (Sections 4.2-4.4). Going back to the original variables, to obtain a solution in the framework of $H^{1}\times L^{2}$, we are forced to apply a trace argument which requires higher regularity of the solution $v$ (Section 4.5). This is why we use the energy method for $v$ in the framework of $H^{2}\times H^{1}$. The restriction $1\leq N\leq 4$ implies that $H^{2}\hookrightarrow L^{q}$ for every $2\leq q<\infty$, which simplifies the energy argument. The blow-up estimate (1.4) is a consequence of an ODE blow-up estimate for the solution of the transformed equation, and the change of variable (Section 4.6).

1.5. Notation

We fix a smooth, even function $\chi:{\mathbb{R}}\to{\mathbb{R}}$ satisfying:

[TABLE]

Let $f(u)=|u|^{p-1}u$ and $F(u)=\int_{0}^{u}f(v)\,dv$. For future reference, we state and justify two Taylor formulas involving the functions $F$ and $f$ (see Introduction of [6] for proofs). Let $\bar{p}=\min(2,p)$. For any $u>0$ and any $v$, it holds

[TABLE]

and

[TABLE]

In the present article, we use multi-variate notation and results from [8]. For $\beta=(\beta_{1},\ldots,\beta_{N})\in{\mathbb{N}}^{N}$ and $x=(x_{1},\ldots,x_{N})\in{\mathbb{R}}^{N}$, we set

[TABLE]

For $\beta,\beta^{\prime}\in{\mathbb{N}}^{N}$, we write $\beta^{\prime}\leq\beta$ provided $\beta_{j}^{\prime}\leq\beta_{j}$, for all $j=1,\ldots,N$. Note that in this case $|\beta-\beta^{\prime}|=|\beta|-|\beta^{\prime}|$. For $\beta^{\prime}\leq\beta$, we denote

[TABLE]

Recall that for two functions $g,h:{\mathbb{R}}^{1+N}\to{\mathbb{R}}$, Leibniz’s formula writes:

[TABLE]

We write $\beta^{\prime}\prec\beta$ provided one of the following holds

• •

$|\beta^{\prime}|<|\beta|$;

• •

$|\beta^{\prime}|=|\beta|$ and $\beta_{1}^{\prime}<\beta_{1}$;

• •

$|\beta^{\prime}|=|\beta|$, $\beta_{1}^{\prime}=\beta_{1}$,…, $\beta_{\ell}^{\prime}=\beta_{\ell}$ and $\beta_{\ell+1}^{\prime}<\beta_{\ell+1}$ for some $1\leq\ell<N$.

We recall the Faa di Bruno formula (see Corollary 2.10 in [8]). Let $n=|\beta|\geq 1$. Then, for functions $f:{\mathbb{R}}\to{\mathbb{R}}$ and $g:{\mathbb{R}}^{1+N}\to{\mathbb{R}}$,

[TABLE]

where

[TABLE]

We will also need to differentiate in space and time, so we define multi-index notation in space-time: ${\boldsymbol{\nu}}=(\alpha,\beta_{1},\ldots,\beta_{N})\in{\mathbb{N}}^{1+N}$, $\beta=(\beta_{1},\ldots,\beta_{N})$, and

[TABLE]

For ${\boldsymbol{\nu}},{\boldsymbol{\nu}}^{\prime}\in{\mathbb{N}}^{1+N}$, we write ${\boldsymbol{\nu}}^{\prime}\leq{\boldsymbol{\nu}}$ provided $\alpha^{\prime}\leq\alpha$ and $\beta_{j}^{\prime}\leq\beta_{j}$, for all $j=1,\ldots,N$. In such a case, we denote

[TABLE]

Then, for two functions $g,h:{\mathbb{R}}^{1+N}\to{\mathbb{R}}$:

[TABLE]

We write ${\boldsymbol{\nu}}^{\prime}\prec{\boldsymbol{\nu}}$ provided one of the following holds

• •

$|{\boldsymbol{\nu}}^{\prime}|<|{\boldsymbol{\nu}}|$;

• •

$|{\boldsymbol{\nu}}^{\prime}|=|{\boldsymbol{\nu}}|$ and $\alpha^{\prime}<\alpha$;

• •

$|{\boldsymbol{\nu}}^{\prime}|=|{\boldsymbol{\nu}}|$, $\alpha^{\prime}=\alpha$ and $\beta_{1}^{\prime}<\beta_{1}$; or

• •

$|{\boldsymbol{\nu}}^{\prime}|=|{\boldsymbol{\nu}}|$, $\alpha^{\prime}=\alpha$, $\beta_{1}^{\prime}=\beta_{1}$,…, $\beta_{\ell}^{\prime}=\beta_{\ell}$ and $\beta_{\ell+1}^{\prime}<\beta_{\ell+1}$ for some $1\ \leq\ell<N$.

Last, we write in this context the Faa di Bruno formula. Let $n=|{\boldsymbol{\nu}}|\geq 1$. Then, for functions $f:{\mathbb{R}}\to{\mathbb{R}}$ and $g:{\mathbb{R}}^{1+N}\to{\mathbb{R}}$,

[TABLE]

where

[TABLE]

2. Blow up ansatz

2.1. Change of variables

Let $\psi\in{\mathcal{C}}^{q_{0}}({\mathbb{R}}^{N},{\mathbb{R}})$, where $q_{0}$ is defined by (1.2), be such that for some $R\geq 2$,

[TABLE]

We perform a change of variable related to $\psi$:

[TABLE]

so that $s>0$ is equivalent to $t<\psi(x)$.

Then, the following holds, for $j=1,\ldots,N$,

[TABLE]

Therefore, the equation (1.1) on $u(t,x)$ rewrites

[TABLE]

In this section, we focus on finding ansatz for this equation under assumption (2.1).

2.2. First blow up ansatz

Let

[TABLE]

where $q_{0}$ is defined by (1.2), and let

[TABLE]

be an integer.

We consider the function $A:{\mathbb{R}}^{N}\to[0,+\infty[$ given by

[TABLE]

It follows that $A$ is of class ${\mathcal{C}}^{k-1}$ and that, for any $\beta\in{\mathbb{N}}^{N}$, with $|\beta|\leq k-1$,

[TABLE]

We define a basic blow up ansatz $V_{0}$, for $s>0$ and $x\in{\mathbb{R}}^{N}$,

[TABLE]

where

[TABLE]

which satisfies $(1-|\nabla\psi|^{2})\partial_{ss}V_{0}=V_{0}^{p}$ on $(0,+\infty)\times{\mathbb{R}}^{N}$. Since the functions $\psi$ and $A$ are of class ${\mathcal{C}}^{q_{0}}$, we remark that the function $V_{0}$ is of class ${\mathcal{C}}^{\infty}$ in the variable $s>0$ and of class ${\mathcal{C}}^{q_{0}-1}$ in the variable $x\in{\mathbb{R}}^{N}$.

In view of (2.2), it is natural to set

[TABLE]

We gather in the next lemma the properties of $V_{0}$ and ${\mathcal{E}}_{0}$.

Lemma 2.1**.**

The function $V_{0}$ satisfies

[TABLE]

Moreover, for any $\alpha\in{\mathbb{N}}$, $\beta\in{\mathbb{N}}^{N}$, $\rho\in{\mathbb{R}}$, $0<s<1$, $x\in{\mathbb{R}}^{N}$, the following hold:

1. (i)

If $0\leq|\beta|\leq q_{0}-1$ and $|x|\leq R$, then

[TABLE] 2. (ii)

If $|\beta|\leq q_{0}-3$ and $|x|\leq R$, then

[TABLE] 3. (iii)

If $|x|>R$, then

[TABLE]

Furthermore if $|x_{0}|<1$, then for any $\sigma>0$,

[TABLE]

Proof.

First, we observe that the function $\kappa$ is constant for $|x|>R$ and satisfies $\kappa\gtrsim 1$, $|\partial_{x}^{\beta}\kappa|\lesssim 1$ on ${\mathbb{R}}^{N}$, for any $|\beta|\leq q_{0}-1$.

Proof of (2.9). This follows from direct computations.

Proof of (2.10). For $0<s<1$ and $|x|\leq R$, one has $0<s+A(x)\lesssim 1$ and thus, $V_{0}\gtrsim 1$. We introduce some notation:

[TABLE]

In particular, $V_{0}(s,x)=\kappa(x)h(W(s,x))$. Let $\alpha\geq 0$. Since $|h^{(\alpha)}(z)|\lesssim|z|^{-\frac{2}{p-1}-\alpha}$, we have

[TABLE]

Let $\beta\in{\mathbb{N}}^{N}$ be such that $1\leq|\beta|\leq q_{0}-1$. Using (1.12), we have

[TABLE]

For $0\leq\beta^{\prime}\leq\beta$, it holds $|\partial_{x}^{\beta-\beta^{\prime}}\kappa|\lesssim 1$. Thus, for $\beta^{\prime}=0$ in the above sum, we have

[TABLE]

For $1\leq|\beta^{\prime}|$, $\beta^{\prime}\leq\beta$, setting $n^{\prime}=|\beta^{\prime}|$ and using (1.13),

[TABLE]

where

[TABLE]

As before, we use for $r\geq 1$, $|h^{(\alpha+r)}(W)|\lesssim W^{-\frac{2}{p-1}-r-\alpha}$. Moreover, using the assumption (2.6) on $A$, we have, for $1\leq|\beta_{\ell}|\leq q_{0}-1$,

[TABLE]

Since $\sum_{\ell=1}^{n^{\prime}}r_{\ell}=r$, $\sum_{\ell=1}^{n^{\prime}}r_{\ell}|\beta_{\ell}|=|\beta^{\prime}|$ and $|\beta^{\prime}|\leq q_{0}-1\leq k-1$, we obtain

[TABLE]

We obtain, for all $0\leq|\beta|\leq q_{0}-1$ and $|x|\leq R$,

[TABLE]

which proves (2.10) for $\rho=1$.

We use the notation ${\boldsymbol{\nu}}=(\alpha,\beta_{1},\ldots,\beta_{N})$ as in the context of formula (1.15). Let $n=|{\boldsymbol{\nu}}|\geq 1$. Then, by (1.15), for $\rho\in{\mathbb{R}}$,

[TABLE]

where

[TABLE]

Using (2.15) and $\sum_{\ell=1}^{n}r_{\ell}=r$, $\sum_{\ell=1}^{n}r_{\ell}\alpha_{\ell}=\alpha$, $\sum_{\ell=1}^{n}r_{\ell}\beta_{\ell}=\beta$ in $P({\boldsymbol{\nu}},r)$, we estimate

[TABLE]

Proof of (2.11). We estimate the three terms in (2.8). It follows from Leibniz’s formula (1.14), the properties of $\psi$, $V_{0}\gtrsim 1$, and estimate (2.10) that, for $|\beta|\leq q_{0}-3$, and $|x|\leq R$,

[TABLE]

Using once more that $V_{0}\gtrsim 1$ for $|x|\leq R$ and $k\geq 1$, these estimates imply (2.11).

Proof of (2.12). It follows from the properties of the functions $\psi$ and $A$ that $V_{0}(s,x)={\kappa_{0}}(s+|x|^{k})^{-\frac{2}{p-1}}$ for any $|x|\geq R$. Estimate (2.12) follows immediately. Then, we have, for any $|x|\geq R$,

[TABLE]

which implies (2.13).

Finally, we prove (2.14). Since $|x_{0}|<1$, we have for $s$ small $|\partial_{t}V_{0}|\gtrsim s^{-\frac{p+1}{p-1}}$, and (2.14) follows. ∎

2.3. Refined blow up ansatz

Starting from $V_{0}$, we define by induction a refined ansatz to the nonlinear wave equation (2.2).

Let $V_{0}$ be defined in (2.7) and let ${\mathcal{E}}_{0}$ be defined in (2.8). Let $s_{0}=1$. For $j\geq 1$, let

[TABLE]

where $\chi_{j}(x)=\chi(A(x)/{r_{j}})$ and $0<r_{j}\leq 1$, $0<s_{j}\leq 1$ are parameters to be defined for each $j=1,\ldots,J$. Since $V_{0}$ is of class ${\mathcal{C}}^{\infty}$ in $s$ and of class ${\mathcal{C}}^{q_{0}-1}$ in $x$, the above expressions make sense as continuous functions for $j$ such that $j\leq J$. This restriction is due to the spatial derivatives in $V_{j}$ in the expression of ${\mathcal{E}}_{j}$.

Lemma 2.2**.**

There exist $0<r_{J}\leq\cdots\leq r_{1}\leq 1$ and $0<s_{J}\leq\cdots\leq s_{1}\leq 1$ such that for any $0\leq j\leq J$, for any $\alpha\in{\mathbb{N}}$, $\beta\in{\mathbb{N}}^{N}$, $0<s\leq s_{j}$, $x\in{\mathbb{R}}^{N}$, the following hold:

1. (i)

If $1\leq j\leq J$, $|\beta|\leq q_{0}-2j-1$ and $|x|\leq R$, then

[TABLE] 2. (ii)

If $1\leq j\leq J$, then

[TABLE] 3. (iii)

If $|\beta|\leq q_{0}-2(j+1)-1$ and $|x|\leq R$, then

[TABLE] 4. (iv)

If $|x|>R$, then

[TABLE]

Remark 2.3**.**

To complete the energy control in Section 3, we need an error estimate of the form $\|{\mathcal{E}}_{J}\|_{L^{2}}\lesssim s^{\frac{2}{p-1}+\delta}$, as in [7] (see (3.28)), as well as an estimate of the form $\|\partial_{s}{\mathcal{E}}_{J}\|_{L^{2}}\lesssim s^{-1+\delta}$ (see the proof of (3.30)), with $\delta>0$. This requires a sufficiently large $J$, see (2.3), and then a sufficiently large $k$. Compared with Lemma 2.3 (see also Remark 2.4) in [7], we need twice as many steps. This is due to the terms depending on $\partial_{s}V_{j}$ in the expression of the error term ${\mathcal{E}}_{j}$. These necessary restrictions have the important consequence that the minimal regularity of the hypersurface that we can consider in Theorem 1.1 depends on $p$, see (2.3).

Proof of Lemma 2.2.

We observe that (2.19), (2.20) and (2.21) for $j=0$ are exactly (2.11), (2.12) and (2.13) in Lemma 2.1. We proceed by induction on $j$: for any $1\leq j\leq J$, we prove that estimate (2.19) for ${\mathcal{E}}_{j-1}$ implies (2.16)–(2.19) for $v_{j}$, $V_{j}$ and ${\mathcal{E}}_{j}$. Let $s_{0}=1$.

Proof of (2.16). Let $1\leq j\leq J$. First, assuming (2.19) for ${\mathcal{E}}_{j-1}$, we show the following estimates related to the two components of $v_{j}$, for $|\beta|\leq q_{0}-2j-1$, $0<s<s_{j-1}$ and $|x|\leq R$,

[TABLE]

Indeed, we have by Leibniz’s formula

[TABLE]

and thus using (2.10) and (2.19) for ${\mathcal{E}}_{j-1}$, we obtain

[TABLE]

For $\alpha=0$, $|\beta|\leq q_{0}-1\leq k-1$ and $1\leq j\leq J$, we note that

[TABLE]

where

[TABLE]

This means that we can integrate this term on $(0,s)$ for $0<s\leq s_{j-1}$. We obtain

[TABLE]

For $\alpha\geq 1$,

[TABLE]

which proves (2.22). Similarly, using Leibniz’s formula, we check the estimate

[TABLE]

In particular, for $\alpha=0$,

[TABLE]

where, using $1\leq j\leq J\leq\frac{2p+2}{p-1}$,

[TABLE]

Thus, by integration on $(s,s_{j-1})$,

[TABLE]

For $\alpha\geq 1$,

[TABLE]

which proves (2.23).

Using estimates (2.10), (2.22), (2.23) and again Leibniz’s formula, we obtain, for all $s\in(0,s_{j-1}]$,

[TABLE]

These estimates implies (2.16) for $v_{j}$ on $(0,s_{j-1}]$.

Proof of (2.17)–(2.18). For $j=1$, we prove (2.17) as a consequence of (2.16). For $2\leq j\leq J$, we prove (2.17) as a consequence of (2.16) for $j$ and (2.17) for $j-1$.

For $|x|>R\geq 1$, (2.6) implies $A(x)\geq 2^{k}\geq 2r_{1}\geq\cdots\geq 2r_{j}$, thus $\chi_{j}=0$ and $V_{j}=V_{0}$.

For $0<s\leq s_{j-1}$ and $|x|<R$, by (2.16) with $\alpha=0$ and $\beta=0$, using the definition of $\chi_{j}$ and the bound $V_{0}\gtrsim 1$, we have

[TABLE]

Choosing $0<r_{j}\leq 1$ and $0<s_{j}\leq s_{j-1}$ sufficiently small, we impose, for $s\in(0,s_{j}]$,

[TABLE]

In the case $j=1$, this proves (2.17). For $j\geq 2$, combining this estimate with (2.17) for $j-1$, we find, for all $s\in(0,s_{j}]$ and $x\in{\mathbb{R}}^{N}$,

[TABLE]

which is (2.17).

To prove (2.18), we note that by (2.16), and using $A\lesssim V_{0}^{-\frac{p-1}{2}}$,

[TABLE]

Proof of (2.19). Note that (2.19) for $j=0$ was already checked. Now, for $1\leq j\leq J$, we prove (2.19) for ${\mathcal{E}}_{j}$ assuming (2.19) for ${\mathcal{E}}_{j-1}$, (2.16) for $v_{j}$ and (2.17) for $V_{j}$. This suffices to complete the induction argument.

By direct computations, we briefly check that the function $v_{j}$ satisfies

[TABLE]

Indeed, we have

[TABLE]

and thus, using (2.9),

[TABLE]

Differentiating in $s$ again, and using (2.9), we obtain

[TABLE]

which is (2.24).

Using (2.24), $V_{j}=V_{j-1}+\chi_{j}v_{j}$ and the definition of ${\mathcal{E}}_{j-1}$, we have

[TABLE]

We estimate each term of the right-hand side above for $|x|\leq R$.

For the first term, recall that for $A\leq r_{j}$, and any $\beta^{\prime}$, $1-\chi_{j}=0$ and $\partial_{x}^{\beta^{\prime}}\chi_{j}=0$. Moreover, for $0<s\leq 1$, for $x$ such that $A(x)>r_{j}$ and $|x|\leq R$, one has $A\approx 1$ and $V_{0}\approx 1$. Thus, using (2.19) for ${\mathcal{E}}_{j-1}$, we find

[TABLE]

Now, we treat the next three terms in the expression of ${\mathcal{E}}_{j}$. By Leibniz’s formula, the properties of $\psi$ and $\chi_{j}$, (2.16) and then $V_{0}\gtrsim 1$, we have, for $0<s\leq s_{j}$ and $|x|<R$,

[TABLE]

we see that these three terms are estimated by $V_{0}^{\frac{p+1}{2}+(-j+\alpha+\frac{1+j+|\beta|}{k})\frac{p-1}{2}}$.

Finally, we estimate $\partial_{s}^{\alpha}\partial_{x}^{\beta}[f(V_{j})-f(V_{j-1})-f^{\prime}(V_{0})\chi_{j}v_{j}]$ using Taylor expansions on $f$ and its derivatives. We start with the case $\alpha=\beta=0$. Recall that by (2.17), we have $0<\frac{3}{4}V_{0}\leq V_{j}\leq\frac{5}{4}V_{0}$. The following Taylor expansions hold:

[TABLE]

and

[TABLE]

These estimates imply

[TABLE]

For $1\leq\ell\leq j$, using (2.16) and next $V_{0}\gtrsim 1$, we have

[TABLE]

Thus, $\left|f(V_{j})-f(V_{j-1})-f^{\prime}(V_{0})\chi_{j}v_{j}\right|\lesssim V_{0}^{\frac{p+1}{2}+(-j+\frac{1+j}{k})\frac{p-1}{2}}$ is proved.

Now, we consider the case $|\alpha|+|\beta|\geq 1$. By the Taylor formula with integral remainder we have for any $V$ and $w$

[TABLE]

Therefore, using the notation ${\boldsymbol{\nu}}=(\alpha,\beta_{1},\ldots,\beta_{N})$, by the Leibniz formula (1.14),

[TABLE]

and, by the Faa di Bruno formula (1.15), for ${\boldsymbol{\nu}}^{\prime}\neq 0$, denoting $n^{\prime}=|{\boldsymbol{\nu}}^{\prime}|$,

[TABLE]

To estimate the term $\partial^{\boldsymbol{\nu}}[f(V_{j})-f(V_{j-1})-f^{\prime}(V_{j-1})\chi_{j}v_{j}]$, we apply these formulas to $V=V_{j-1}$ and $w=\chi_{j}v_{j}$. First, for ${\boldsymbol{\nu}}^{\prime}\leq{\boldsymbol{\nu}}$, using (2.16) and the properties of $\chi$, we obtain

[TABLE]

Thus, for ${\boldsymbol{\nu}}^{\prime}=0$ and $\theta\in[0,1]$, from (2.17), we obtain

[TABLE]

Second, for ${\boldsymbol{\nu}}^{\prime}\neq 0$, ${\boldsymbol{\nu}}^{\prime}\leq{\boldsymbol{\nu}}$ and $\theta\in[0,1]$, from formula (2.25), using (2.10) and (2.17), we have (the definition of $P({\boldsymbol{\nu}}^{\prime},r)$ implies $\sum_{\ell=1}^{n^{\prime}}r_{\ell}=r$, $\sum_{\ell=1}^{n^{\prime}}r_{\ell}{\boldsymbol{\nu}}_{\ell}={\boldsymbol{\nu}}^{\prime}$)

[TABLE]

Thus, we have proved

[TABLE]

and so by integration in $\theta\in[0,1]$,

[TABLE]

We now estimate $\partial^{\boldsymbol{\nu}}[f(V_{j})-f(V_{j-1})-f^{\prime}(V_{0})\chi_{j}v_{j}]$. For any $V,W,w$, we have

[TABLE]

and thus

[TABLE]

Moreover, for ${\boldsymbol{\nu}}^{\prime}\neq 0$, formula (2.25) (with $V$ replaced by $W$, and $w$ by $V-W$) yields

[TABLE]

To estimate the term $\partial^{\boldsymbol{\nu}}[\chi_{j}v_{j}(f^{\prime}(V_{j-1})-f^{\prime}(V_{0}))]$, we apply these formulas to $V=V_{j-1}$, $W=V_{0}$ and $w=\chi_{j}v_{j}$.

For ${\boldsymbol{\nu}}^{\prime}\leq{\boldsymbol{\nu}}$, using (2.16) and Leibniz’s formula, we have, for $1\leq\ell\leq j-1$,

[TABLE]

For ${\boldsymbol{\nu}}^{\prime}=0$ and $\theta\in[0,1]$, from (2.17), we obtain

[TABLE]

Second, for ${\boldsymbol{\nu}}^{\prime}\neq 0$, ${\boldsymbol{\nu}}^{\prime}\leq{\boldsymbol{\nu}}$ and $\theta\in[0,1]$, by formula (2.25), using (2.10), (2.16) and (2.17), we have

[TABLE]

Thus, we obtain

[TABLE]

Integrating in $\theta\in[0,1]$ and summing in ${\boldsymbol{\nu}}^{\prime}\leq{\boldsymbol{\nu}}$, we obtain

[TABLE]

Combining (2.26) and (2.27), we have proved for $s\in(0,s_{j}]$, $|x|\leq R$,

[TABLE]

In conclusion, we have estimated all terms in the expression of ${\mathcal{E}}_{j}$ and (2.19) for $j$ is proved.

Proof of (2.20)–(2.21). For $|x|>R\geq 2$, (2.6) implies $A(x)\geq 2^{k}\geq 2r_{1}\geq\cdots\geq 2r_{j}$, thus $\chi_{j}=0$ and $V_{j}=V_{0}$, ${\mathcal{E}}_{j}={\mathcal{E}}_{0}$. Thus, (2.20)–(2.21) follow from (2.12)–(2.13). ∎

3. Construction of a solution of the transformed equation $\eqref{eq:v}$

Let the function $\chi$ be given by (1.7), let $\psi\in{\mathcal{C}}^{q_{0}}({\mathbb{R}}^{N},{\mathbb{R}})$, where $q_{0}$ is defined by (1.2), satisfy (2.1), let $J$, $q_{0}$ and $k$ be as in (2.3)-(2.4). Set

[TABLE]

and impose the following additional condition on $k$

[TABLE]

Recall that $A:{\mathbb{R}}^{N}\to[0,+\infty[$ is defined by (2.5), and let $V_{J}$ be defined as in Section 2.3.

Our main result of this section is the following.

Proposition 3.1**.**

Assume that

[TABLE]

There exist $0<\delta_{0}<1$ and a function

[TABLE]

which is a solution of (2.2) in $C((0,\delta_{0}),L^{2}({\mathbb{R}}^{N}))$, and which satisfies

[TABLE]

for all $0<s<\delta_{0}$, with $\lambda$ given by (3.1). In addition, there exist a constant $C$ and a function $g\in L^{\infty}((0,\delta_{0}),H^{1}({\mathbb{R}}^{N}))$ such that

[TABLE]

a.e. on $(0,\delta_{0})\times{\mathbb{R}}^{N}$.

We construct the solution $v$ of Proposition 3.1 by a compactness argument. For any $n$ large, let $S_{n}=\frac{1}{n}<s_{J}$ and

[TABLE]

We let $n$ be sufficiently large so that $B_{n}\geq 1$, and we define the function $f_{n}:{\mathbb{R}}\to[0,\infty)$ by

[TABLE]

Let $F_{n}(u)=\int_{0}^{u}f_{n}(w)dw$. Note that Taylor’s estimates such as (1.8)–(1.11) still hold for $F_{n}$ and $f_{n}$ with constants independent of $n$. We will refer to these inequalities for $F_{n}$ and $f_{n}$ with the same numbers (1.8), (1.9) and (1.11). In this proof, any implicit constant related the symbol $\lesssim$ is independent of $n$.

We define the sequence of solution $v_{n}$ of

[TABLE]

The nonlinearity $f_{n}$ being globally Lipschitz, the existence of a global solution $(v_{n},\partial_{s}v_{n})$ in $H^{2}\times H^{1}$ is a consequence of standard arguments from semigroups theory, see Appendix A, and in particular Section A.4.

We set, for all $s\in[S_{n},s_{J}]$,

[TABLE]

thus $(w_{n},\partial_{s}w_{n})\in\mathcal{C}([S_{n},s_{J}],H^{2}({\mathbb{R}}^{N})\times H^{1}({\mathbb{R}}^{N}))\cap\mathcal{C}^{1}([S_{n},s_{J}],H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N}))$. The crucial step in the proof of Proposition 3.1 is the following estimate.

Proposition 3.2**.**

There exist $C>0$, $n_{0}>0$ and $0<\delta_{0}<1$ such that

[TABLE]

for all $n\geq n_{0}$ and $s\in[S_{n},S_{n}+\delta_{0}]$.

Proof.

We fix $n\geq n_{0}$ large, and we denote $w_{n}$ simply by $w$ in this proof. By (3.7) and the definition of ${\mathcal{E}}_{J}$, $w$ satisfies the equation

[TABLE]

We define the auxiliary function $Q$ as follows

[TABLE]

where, by abuse of notation, we denote $\chi(x)=\chi(|x|)$. Note that $Q\gtrsim 1$. We make the following preliminary observation

[TABLE]

Thus, setting

[TABLE]

(by the definition of $Q$ and $V_{0}$, we expect $G$ to be small in some sense), we rewrite the equation of $w$ as follows

[TABLE]

The nonlinear term $f_{n}(V_{J}+w)-f_{n}(V_{J})-f_{n}^{\prime}(V_{0})w$ is mostly quadratic in $w$ (some linear terms in $w$ remain but they are also small in $V_{J}-V_{0}$), which is an important gain with respect to the previous formulation.

We define the following energy functional related to the above formulation of the equation of $w$

[TABLE]

We also define a weighted norm related to the above functional

[TABLE]

Since we may be dealing with $H^{1}\times L^{2}$ supercritical nonlinearities (but $H^{2}\times H^{1}$ subcritical by the condition $1\leq N\leq 4$), we need higher order energy functionals. We set

[TABLE]

and

[TABLE]

For future reference, we establish two estimates on $\partial_{s}Q$ and $\nabla Q$. By the expression of $V_{0}$ in (2.9), we have

[TABLE]

Thus, since $Q^{\frac{p-1}{2(p+1)}}\lesssim s^{-1}$,

[TABLE]

Similarly, by (2.10),

[TABLE]

and

[TABLE]

Step 1. Coercivity. We claim the following estimates.

Lemma 3.3**.**

It holds

[TABLE]

For $0<\delta\leq s_{J}$ and $0<\omega\leq 1$ sufficiently small, for $n$ large, if ${\mathcal{N}}\leq\omega$ and ${\mathcal{M}}\leq\omega$, then

[TABLE]

Proof.

First, we prove the following estimates. For any $\rho\geq 0$, the following holds on $[S_{n},\delta_{0}]$,

[TABLE]

We have, using (3.13),

[TABLE]

This proves (3.17) and the proof of (3.18) is similar. Moreover, using (3.12),

[TABLE]

which proves (3.19); the proof of (3.20) is similar.

We prove (3.15). The inequality $\|w\|_{L^{2}}\lesssim{\mathcal{N}}$ is obvious. Next, (3.19) with $\rho=1$ and $Q^{\frac{p-1}{p+1}}\lesssim s^{-2}$ show that $\int|\partial_{s}w|^{2}\lesssim\int Q|\partial_{s}w|^{2}\lesssim{\mathcal{N}}^{2}$. Since $\|\nabla\psi\|_{L^{\infty}}\leq\frac{1}{2}$ (from (3.3)), it follows (using $\|w\|_{H^{2}}\lesssim\|\Delta w\|_{L^{2}}+\|w\|_{L^{2}}$) that ${\mathcal{M}}^{2}\lesssim{\mathcal{N}}^{2}+{\mathcal{K}}$, which is (3.15).

Last, we prove (3.16). Let

[TABLE]

The triangle inequality and the Taylor inequality (1.8) yield

[TABLE]

where

[TABLE]

From (2.17), $V_{J}\lesssim V_{0}$ and $|V_{J}-V_{0}|\lesssim(1+V_{0})^{-\frac{p-1}{4}}V_{0}\lesssim Q^{-\frac{p-1}{4(p+1)}}V_{0}$. Moreover, $V_{0}^{p+1}\leq Q$. Thus,

[TABLE]

and so

[TABLE]

For the first term, we prove the following general estimate: for any $0<\zeta\leq 1$,

[TABLE]

Indeed, using Hölder’s inequality and the embedding $H^{2}({\mathbb{R}}^{N})\hookrightarrow L^{q}({\mathbb{R}}^{N})$ for $2\leq q<\infty$ (recall that $N\leq 4$),

[TABLE]

In particular, from (3.24), it holds

[TABLE]

In the case $1<p\leq 2$, one has $\bar{p}=p$ and the second term is identical to the first one. In the case $p>\bar{p}=2$, the second term is estimated as follows. Using the inequality $|w|^{3}\leq aw^{2}+a^{-(p-2)}|w|^{p+1}$ with $a=\varepsilon Q^{\frac{1}{p-1}}$, $\varepsilon>0$ to be chosen later, and the estimate $Q^{\frac{p-1}{p+1}}\lesssim s^{-2}$, we see that

[TABLE]

and so, using (3.24)

[TABLE]

Last, since $Q^{\frac{p-1}{2(p+1)}}\lesssim s^{-1}$, we observe that

[TABLE]

In conclusion, we have obtained, for $\mathcal{N}\leq\omega$, $\mathcal{M}\leq\omega$, $S_{n}\leq s\leq\delta$,

[TABLE]

which, combined with (3.15), implies that for $\delta>0$ and $\omega>0$ small enough, it holds $2{\mathcal{H}}+{\mathcal{K}}\geq{\mathcal{N}}^{2}$ on $[S_{n},\delta_{0}]$. (Recall that $1-|\nabla\psi|^{2}\geq\frac{3}{4}$ by (3.3) and $\lambda\leq\frac{1}{2}$.) ∎

Step 2. Energy control. We claim that there exist $C>0$ such that

[TABLE]

provided ${\mathcal{N}}\leq\omega$ and ${\mathcal{M}}\leq\omega$ with $\omega$ sufficiently small.

Proof of (3.26). We compute $\frac{d{\mathcal{H}}}{ds}$:

[TABLE]

First, we remark the negative contribution of $I_{2}$. Since $\partial_{s}Q\leq 0$ by (3.11), we have

[TABLE]

Second, we compute $I_{1}$ using the equation (3.10) of $w$

[TABLE]

For $I_{7}$, we first observe that

[TABLE]

Second, by integration by parts,

[TABLE]

By the definition of ${\mathcal{N}}$, we estimate

[TABLE]

Using (3.13), we also have

[TABLE]

Now, by the expressions of $Q$ and $V_{0}$, we have

[TABLE]

and thus

[TABLE]

Similarly, using (3.12), (3.13), (3.14)

[TABLE]

and

[TABLE]

Using the same estimates and then (3.19), we finish estimating $I_{7}$ as follows

[TABLE]

Thus, for some constant $C>0$, using (3.3),

[TABLE]

Next, integrating by parts, using the identities

[TABLE]

and integrating again by parts, we find

[TABLE]

By (3.13) and the definition of ${\mathcal{N}}$,

[TABLE]

Similarly, using (3.13) and (3.14), we have $|\Delta(Q^{\frac{1}{2}})Q^{\frac{3}{2}}|\lesssim|\nabla Q|^{2}+|\Delta Q|Q\lesssim s^{-\frac{2}{k}}Q^{2}$, and thus

[TABLE]

For $I_{9}$, we start by an estimate of $G=f_{n}^{\prime}(V_{0})Q^{\frac{1}{2}}-(1-|\nabla\psi|^{2})\partial_{ss}(Q^{\frac{1}{2}})$. By the definition of $Q=(1-\chi+V_{0})^{p+1}$ and (2.9), we observe

[TABLE]

Thus,

[TABLE]

For $|x|>1$, we have $V_{0}\lesssim 1$ and $Q\lesssim 1$; since also $Q\gtrsim 1$, we see that $\|G\|_{L^{\infty}}\lesssim 1$. Therefore,

[TABLE]

For $I_{10}$, by the Cauchy-Schwarz inequality

[TABLE]

and we need only estimate $\|Q^{\frac{1}{2}}{\mathcal{E}}_{J}\|_{L^{2}}$. From (2.21), for $|x|\geq R$, we have

[TABLE]

Since $1\leq N\leq 4$, this implies $\|Q^{\frac{1}{2}}{\mathcal{E}}_{J}\|_{{L^{2}(|x|>R})}\lesssim 1$.

Next, from (2.19), for $|x|\leq R$, we have

[TABLE]

Recall that by (3.1),

[TABLE]

and that (3.2) is equivalent to

[TABLE]

Thus, for $|x|\leq R$,

[TABLE]

It follows that

[TABLE]

For this term, we have obtained

[TABLE]

Finally, by the Cauchy-Schwarz inequality,

[TABLE]

Using (3.27), we obtain

[TABLE]

In conclusion for $I_{1}+I_{2}$, we find

[TABLE]

To continue with the proof of (3.26), we estimate the term $I_{3}$. To that end, recall that $\bar{p}=\min(2,p)$. First, by (3.21)–(3.23) and (3.12)

[TABLE]

Using (3.24), the first term is controled as follows

[TABLE]

In the case $1<p\leq 2$, one has $\bar{p}=p$ and the second term is identical to the first one. In the case $p>\bar{p}=2$, using $Q^{\frac{p-1}{2(p+1)}}\lesssim s^{-1}$ and (3.25),

[TABLE]

where $\varepsilon>0$ is to be chosen. Last, we observe that $Q^{\frac{9p-1}{4(p+1)}}w^{2}\lesssim s^{-\frac{5}{2}}Qw^{2}$, and thus

[TABLE]

In conclusion, we have proved

[TABLE]

We proceed similarly for $I_{4}$. Indeed, setting

[TABLE]

by (1.9) and Taylor’s inequality,

[TABLE]

Using (3.29), we conclude that

[TABLE]

Now, we estimate $I_{5}$ and we set

[TABLE]

By the triangle inequality, Taylor inequality (1.11), $|\partial_{s}V_{0}|\lesssim V_{0}^{\frac{p+1}{2}}$ (see (2.9)), we have

[TABLE]

Using (3.29), we conclude that

[TABLE]

Finally, we estimate $I_{6}$ and we set

[TABLE]

By the triangle inequality and Taylor’s inequality (1.11)

[TABLE]

Using (2.18), $V_{0}\lesssim Q^{\frac{1}{p+1}}$, $Q\gtrsim 1$, $Q\lesssim s^{-\frac{2(p+1)}{p-1}}$ and $k\geq 1$, we obtain

[TABLE]

Using (3.29) and $k\geq 2$, we conclude that

[TABLE]

Choosing $\varepsilon\leq\frac{\lambda}{16}$, then $\omega$ sufficiently small, and collecting the above estimates, we have proved (3.26).

Step 3. Higher order energy terms. We claim that for any $\ell=0,1,\ldots,N$,

[TABLE]

Differentiating (3.9) with respect to $s$, setting $z_{0}=\partial_{s}w$, we have

[TABLE]

Differentiating ${\mathcal{K}}_{0}=\int(1-|\nabla\psi|^{2})(\partial_{s}z_{0})^{2}+|\nabla z_{0}|^{2}$, we find from (3.31) and integration by parts

[TABLE]

First, by the Cauchy-Schwarz inequality

[TABLE]

From

[TABLE]

and then (3.19) with $\rho=1$, we have (recall that $1\leq N\leq 4$ and thus $H^{2}({\mathbb{R}}^{N})\hookrightarrow L^{q}({\mathbb{R}}^{N})$ for all $q\geq 2$, and $H^{1}({\mathbb{R}}^{N})\hookrightarrow L^{q}({\mathbb{R}}^{N})$ for all $2\leq q\leq 4$)

[TABLE]

Thus, using also $-\frac{p-1}{p}\geq-1+\lambda$ (since $\lambda\leq\frac{1}{p}$) we have

[TABLE]

Second, using (2.10) and (2.18), $|\partial_{s}V_{J}|\lesssim V_{0}^{1+\frac{p-1}{2}}+V_{0}^{1+\frac{p-1}{2k}}\lesssim V_{0}Q^{\frac{p-1}{2(p+1)}}$, so that Taylor’s inequality (1.10) yields

[TABLE]

We have

[TABLE]

Moreover, since $\frac{2(p-2)}{p+1}\leq\frac{(2p-1)(p-1)}{p(p+1)}$, $Q\gtrsim 1$, $Q\lesssim s^{-\frac{2(p+1)}{p-1}}$, and $\lambda\leq\frac{1}{p}$, we have

[TABLE]

Thus,

[TABLE]

Third, from (2.21), for $|x|\geq R$, $|\partial_{s}{\mathcal{E}}_{J}|\lesssim|x|^{-\frac{k(p+1)}{p-1}-2}$ and thus, since $1\leq N\leq 4$, $\|\partial_{s}{\mathcal{E}}_{J}\|_{L^{2}(|x|\geq R)}\lesssim 1$. Now, from (2.19), for $|x|\leq R$,

[TABLE]

and thus, following the proof of (3.28), we have $\|\partial_{s}{\mathcal{E}}_{J}\|_{L^{2}(|x|\leq R)}\lesssim s^{-1+\lambda}$. Thus,

[TABLE]

The above estimates prove (3.30) for ${\mathcal{K}}_{0}$.

We now prove (3.30) for $\ell\in\{1,\dots,N\}$. Differentiating (3.9) with respect to $x_{\ell}$, setting $z_{\ell}=\partial_{x_{\ell}}w$, we have

[TABLE]

Differentiating ${\mathcal{K}}_{\ell}=\int(1-|\nabla\psi|^{2})(\partial_{s}z_{\ell})^{2}+|\nabla z_{\ell}|^{2}$, we find from (3.32) and integration by parts

[TABLE]

The term $I_{15}$ is estimated exactly like $I_{12}$. Next, it follows from (2.10), (2.16), (2.20) and the properties of $\chi$ that $|\partial_{x_{\ell}}V_{J}|\lesssim V_{0}^{1+\frac{p-1}{2k}}$, so that $I_{16}$ is estimated like $I_{13}$.

Moreover, from (2.21), for $|x|\geq R$, $|\partial_{x_{\ell}}{\mathcal{E}}_{J}|\lesssim|x|^{-3}$ and thus, since $1\leq N\leq 4$, $\|\partial_{s}{\mathcal{E}}_{J}\|_{L^{2}(|x|\geq R)}\lesssim 1$. Now, from (2.19), for $|x|\leq R$,

[TABLE]

and thus, following the proof of (3.28), we have $\|\partial_{s}{\mathcal{E}}_{J}\|_{L^{2}(|x|\leq R)}\lesssim s^{-1+\lambda}$. Thus we see that $I_{17}$ is estimated like $I_{14}$.

Finally,

[TABLE]

so that $I_{18}\lesssim{\mathcal{M}}^{2}$. Therefore, the estimate (3.30) holds for $\ell\in\{1,\dots,N\}$.

Step 4. Conclusion. Since ${\mathcal{H}}(S_{n})={\mathcal{K}}(S_{n})=0$, the following is well-defined

[TABLE]

and by continuity, $S_{n}^{\star}\in(S_{n},\delta]$. It follows from (3.26), (3.30), (3.15), and $\lambda\leq\frac{1}{2}$ that

[TABLE]

for some constant $C>0$ independent of $\delta$. By the definition of $S_{n}^{\star}$, we deduce that

[TABLE]

for some constant $C>0$ independent of $\delta$. We fix $0<\delta_{0}\leq\delta$ such that

[TABLE]

This gives, for all $S_{n}\leq s\leq\min(S_{n}^{\star},\delta_{0})$, $\frac{d}{dt}({\mathcal{H}}+{\mathcal{K}})\leq\frac{\lambda}{3}s^{-1+\lambda}$.

By integration, using ${\mathcal{H}}(s_{n})={\mathcal{K}}(s_{n})=0$, we find for $S_{n}\leq s\leq\min(S_{n}^{\star},\delta_{0})$,

[TABLE]

Thus, from (3.16), it holds, for $S_{n}\leq s\leq\min(S_{n}^{\star},\delta_{0})$,

[TABLE]

It follows from (3.15) and the definition of $S_{n}^{\star}$ that $S_{n}^{\star}\geq\delta_{0}$ and so, for all $s\in[S_{n},\delta_{0}]$,

[TABLE]

This completes the proof of the proposition. ∎

Proof of Proposition 3.1.

We set

[TABLE]

From Proposition 3.2, there exist $C>0$, $n_{0}>0$ and $0<\delta_{0}<1$ such that

[TABLE]

for all $n\geq n_{0}$ and $s\in[0,\delta_{0}]$. Moreover, from (3.9),

[TABLE]

It follows from estimate (3.33) that there exist a subsequence of $(\eta_{n})$ (still denoted by $(\eta_{n})$) and a map $\eta\in L^{\infty}((0,\delta_{0}),H^{2}({\mathbb{R}}^{N}))\cap W^{1,\infty}((0,\delta_{0}),H^{1}({\mathbb{R}}^{N}))\cap W^{2,\infty}((0,\delta_{0}),L^{2}({\mathbb{R}}^{N}))$ such that

[TABLE]

It is then easy to pass to the limit in (3.34), and it follows that

[TABLE]

in $L^{\infty}((0,\delta_{0}),L^{2}({\mathbb{R}}^{N}))$. Therefore, setting

[TABLE]

it holds

[TABLE]

and, using the definition of ${\mathcal{E}}_{J}$, we see that $v$ is a solution of equation (2.2) in $L^{\infty}_{\mathrm{loc}}((0,\delta_{0}),L^{2}({\mathbb{R}}^{N}))$. The estimate (3.5) follows by letting $n\to\infty$ in (3.8) and using (3.38) and (3.39). We now prove that $v$ satisfies (3.4). By standard semigroup theory (see Section A.3) it suffices to prove that $|v|^{p-1}v\in C((0,\delta_{0}),H^{1}({\mathbb{R}}^{N}))$. Since by (3.40) $v\in C((0,\delta_{0}),H^{2-\eta}({\mathbb{R}}^{N}))$ for every $\eta>0$, and $N\leq 4$, we have by Sobolev’s embeddings $v\in C((0,\delta_{0}),W^{1,q}({\mathbb{R}}^{N}))$ for all $2\leq q<4$ and $|v|^{p-1}\in C((0,\delta_{0}),L^{r}({\mathbb{R}}^{N}))$ for $\max\{1,\frac{2}{p-1}\}\leq r<\infty$. Choosing for instance $q=\frac{4(p+1)}{p+3}$ and $r=\frac{4(p+1)}{p-1}$ yields $|v|^{p-1}v\in C((0,\delta_{0}),H^{1}({\mathbb{R}}^{N}))$.

Finally, we prove (3.6). We write

[TABLE]

On the other hand, $V_{0}^{1+\frac{p-1}{2}}\lesssim|\partial_{s}V_{0}|$, so that $V_{0}^{1+\frac{p-1}{2k}}\lesssim|\partial_{s}V_{0}|^{1-\frac{(p-1)(k-1)}{k(p+1)}}$. Therefore, given any $\eta>0$, there exists a constant $C_{\eta}$ such that

[TABLE]

Since $|\partial_{s}(V_{j}-V_{0})|\lesssim V_{0}^{1+\frac{p-1}{2k}}$ by (2.18), we see that there exists a constant $C$ such that

[TABLE]

Next, we write

[TABLE]

It follows from (2.20) that $|\nabla V_{J}|\lesssim 1$ for $|x|\geq R$. For $|x|<R$, by (2.16) and $k\geq 2$, $|\nabla v_{j}|\lesssim V_{0}$ for $j\geq 1$; and $|\nabla V_{0}|\lesssim V_{0}^{1+\frac{p-1}{2k}}$ by (2.10). Using again (3.41), we conclude that

[TABLE]

Since $|\partial_{s}(v-V_{J})|+|\nabla(v-V_{J})|\in L^{\infty}((0,\delta_{0}),H^{1}({\mathbb{R}}^{N}))$ by (3.5), the lower estimate (3.6) follows from (3.42) and (3.43). ∎

4. Proof of Theorem 1.1

In this section, we use the following notation. We let $\{\mathbf{e}_{k};k=1,\ldots,N\}$ be the canonical basis of ${\mathbb{R}}^{N}$. If $N\geq 2$, then for $x\in{\mathbb{R}}^{N}$, we denote $x=(x_{1},x_{2},\ldots,x_{N})$ and $\bar{x}=(x_{2},\ldots,x_{N})$. We set $\bar{\Delta}u=\sum_{k=2}^{N}\partial_{x_{k}x_{k}}u$. If $N=1$, we ignore $\bar{x}$ and $\bar{\Delta}$.

4.1. Cut-off of the local hypersurface

Let $\varphi$ be a function satisfying (1.3) (see statement of Theorem 1.1). Without loss of generality, by the invariance by rotation of equation (1.1), we assume that

[TABLE]

(For dimension $1$, the reduction is done by possibly changing $x\mapsto-x$.) For a positive real $r<1$ small to be defined later, set

[TABLE]

On the one hand, from this definition and the properties of $\chi$, it holds

[TABLE]

On the other hand, from $\varphi(0)=0$ and $\nabla\varphi(0)=\ell\mathbf{e}_{1}$, there exists a constant $C>1$ such that for $|x|<1$, it holds $|\varphi(x)-\ell x_{1}|\leq C|x|^{2}$ and $|\nabla\varphi(x)-\ell\mathbf{e}_{1}|\leq C|x|$. In particular, since

[TABLE]

it holds on ${\mathbb{R}}^{N}$,

[TABLE]

We fix $r>0$ small enough so that

[TABLE]

The first constraint on $\widetilde{\varphi}$ is related to assumption (3.3) in Proposition 3.2, and the second implies

[TABLE]

4.2. Construction of the function $\psi$

We claim that for any $y\in{\mathbb{R}}^{N}$, there exists $X_{1}(y)\in{\mathbb{R}}$ such that

[TABLE]

(As observed before, we ignore $\bar{y}$ in dimension $1$.) To prove the claim, we define

[TABLE]

and we compute, using (4.3),

[TABLE]

and

[TABLE]

Thus, for fixed $\bar{y}\in{\mathbb{R}}^{N-1}$, the function $x_{1}\in{\mathbb{R}}\mapsto\widetilde{\Phi}_{\bar{y}}(x_{1})=:\Phi(x_{1},\bar{y})\in{\mathbb{R}}$ is increasing and surjective. It has an inverse function $\widetilde{\Phi}_{\bar{y}}^{-1}$ on ${\mathbb{R}}$, which is also (strictly) increasing, and we set $X_{1}(y_{1},\bar{y})=\widetilde{\Phi}_{\bar{y}}^{-1}(y_{1})$ for $y_{1}\in{\mathbb{R}}$. Setting $X_{1}(y)=X_{1}(y_{1},\bar{y})$, we have proved the claim. Note that

[TABLE]

so that by (4.5)

[TABLE]

for all $x\in{\mathbb{R}}^{N}$. Moreover, it follows from (4.6)-(4.7) that

[TABLE]

on ${\mathbb{R}}^{N}$. Setting $X(y)=(X_{1}(y),\bar{y})$, it holds

[TABLE]

Moreover, using (4.9), we see that that

[TABLE]

For all $y\in{\mathbb{R}}^{N}$, we define the function $\psi:{\mathbb{R}}\to{\mathbb{R}}$ by

[TABLE]

Equivalently, the functions $\psi$ and $\widetilde{\varphi}$ are uniquely related by the following relation on ${\mathbb{R}}^{N}$:

[TABLE]

We check that $\psi$ is of class $\mathcal{C}^{q_{0}}$ where $q_{0}$ is defined in (1.2), and satisfies the assumptions (2.1) and (3.3).

First, since $\varphi$ is of class $\mathcal{C}^{q_{0}}$ and $\chi$ is of class $\mathcal{C}^{\infty}$, it follows from their definitions that $\widetilde{\varphi}$ and then the functions $X$ and $\psi$ are of class $\mathcal{C}^{q_{0}}$ in ${\mathbb{R}}^{N}$. Since $\varphi(0)=\widetilde{\varphi}(0)=0$, from (4.13), we also have $\psi(0)=0$.

Second, from (4.1), it follows that $\widetilde{\varphi}(x)=\ell x_{1}$ for any $|x|>2r$. From (4.11) and (4.12), we see that $\psi(y)=0$ for $|y|$ large.

Last, we estimate $|\nabla\psi|$. From (4.13)

[TABLE]

and for $j\neq 1$,

[TABLE]

It follows from (4.3) that $|1-\ell\partial_{x_{1}}\widetilde{\varphi}(x)|\geq 1-\ell$, so that (4.14) and (4.2) yield

[TABLE]

In particular, we see that $\|\partial_{y_{1}}\psi\|_{L^{\infty}}\leq 1$. Since $\|\partial_{x_{j}}\widetilde{\varphi}\|_{L^{\infty}}\leq(1-\ell)\frac{\lambda}{8}\frac{p-1}{p+1}$ by (4.2), we deduce from (4.15) that

[TABLE]

so that (3.3) is proved.

4.3. Definition of an appropriate solution of the transformed equation

We assume (2.3), (2.4), (3.1), (3.2) and we consider the function $\psi$ defined in (4.12)-(4.13). Note that $\psi$ is of class $\mathcal{C}^{q_{0}}$ where $q_{0}$ is defined in (1.2), and satisfies the assumptions (2.1) and (3.3). Let the function $A$ be given by (2.5). We consider the solution $v\in C((0,\delta_{0}),H^{2}({\mathbb{R}}^{N}))\cap C^{1}((0,\delta_{0}),H^{1}({\mathbb{R}}^{N}))\cap C^{2}((0,\delta_{0}),L^{2}({\mathbb{R}}^{N}))$ of (2.2) given by Proposition 3.1.

4.4. Returning to the original variable

Let

[TABLE]

Recall (see (4.1) and (4.3)) that $\widetilde{\varphi}(0)=0$ and $|\nabla\widetilde{\varphi}|\leq\frac{\ell+1}{2}$, so that $|\widetilde{\varphi}(x)|\leq\frac{\ell+1}{2}|x|$. Thus we see that

[TABLE]

It follows that the space-time region

[TABLE]

is an influence domain in the sense of §1.2. (See Figure 1.) Moreover, let $|x|\leq\frac{\varepsilon_{0}}{2}$. We have $\widetilde{\varphi}(x)\leq\frac{\varepsilon_{0}}{2}$. Therefore, if $0\leq t<\tau_{0}+\widetilde{\varphi}(x)$, then $t<\tau_{0}+\frac{\varepsilon_{0}}{2}$ so that $|x|<\tau_{0}+\varepsilon_{0}-t$. It follows that

[TABLE]

Given $0\leq\ell<1$ and $\tau_{0}\in{\mathbb{R}}$, we define the Lorentz transform $\Lambda_{\ell,\tau_{0}}:{\mathbb{R}}^{1+N}\to{\mathbb{R}}^{1+N}$ by

[TABLE]

It is well known that $\Lambda_{\ell,\tau_{0}}$ is a $\mathcal{C}^{\infty}$ diffeomorphism with Jacobian determinant $|\det J_{\Lambda_{\ell,\tau_{0}}}|=1$. We also define the transformation $\Lambda_{\psi}:{\mathbb{R}}^{1+N}\to{\mathbb{R}}^{1+N}$ by

[TABLE]

Since $\psi$ is of class $\mathcal{C}^{q_{0}}$ where $q_{0}$ is defined in (1.2) (see §4.2), it follows easily that $\Lambda_{\psi}$ is a diffeomorphism of class $\mathcal{C}^{q_{0}}$. Moreover, $|\det J_{\Lambda_{\psi}}|=1$. We define the map $\Lambda:{\mathbb{R}}^{1+N}\to{\mathbb{R}}^{1+N}$ as the composition of the above two maps, i.e.

[TABLE]

The map $\Lambda$ has the following expression

[TABLE]

and it follows that $\Lambda:{\mathbb{R}}^{1+N}\to{\mathbb{R}}^{1+N}$ is a diffeomorphism of class $\mathcal{C}^{q_{0}}$ and that $|\det J_{\Lambda}|=1$.

We prove that

[TABLE]

and that

[TABLE]

In the case where $\ell=0$, by (4.4), we have $X(y)=y$ and thus by (4.12), $\psi(y)=\widetilde{\varphi}(y)$. Thus in this case,

[TABLE]

Property (4.20) follows. Moreover, $s\leq\widetilde{\varphi}(x)+\tau_{0}\leq|x|+\tau_{0}\leq 2\tau_{0}+\varepsilon_{0}\leq 3\tau_{0}<\frac{\delta_{0}}{2}$ by (4.16). Thus (4.21) is proved in this case.

In the case where $\ell\neq 0$, we observe that from (4.12),

[TABLE]

Using (4.10), we replace $\widetilde{\varphi}(X(y))=\frac{1}{\ell}(X_{1}(y)-(1-\ell^{2})^{\frac{1}{2}}y_{1})$ so that

[TABLE]

Recall that by (4.8), we have x_{1}=X_{1}\Big{(}\frac{x_{1}-\ell\widetilde{\varphi}(x)}{(1-\ell^{2})^{\frac{1}{2}}},\bar{x}\Big{)}, which means that

[TABLE]

hence, using (4.9),

[TABLE]

on ${\mathbb{R}}^{1+N}$. Thus we see that $s>0$ is equivalent to $t<\tau_{0}+\widetilde{\varphi}(x)$, i.e. (4.20) holds. Moreover, by (4.24), we have on $\mathcal{T}$

[TABLE]

Using (4.16), we see that $\tau_{0}+\widetilde{\varphi}(x)-t<\widetilde{\varphi}(x)+\tau_{0}\leq 3\tau_{0}$, so that $s<\frac{\delta_{0}}{2}$. Thus (4.21) is proved in all cases.

We now set

[TABLE]

We refer to [18], Exercise 10.7.c for a similar use of the Lorentz transform. Note that by (4.21), $u$ is well defined.

Let $\omega$ be an open subset of ${\mathbb{R}}^{N}$ and let $0\leq a<b$. Suppose that $[a,b]\times\overline{\omega}\subset\mathcal{T}$. We claim that

[TABLE]

Since $[a,b]\times\overline{\omega}$ is a compact subset of ${\mathbb{R}}^{1+N}$, it follows that $\Lambda([a,b]\times\overline{\omega})$ is a compact subset of ${\mathbb{R}}^{1+N}$. Moreover, it follows from (4.20)-(4.21) that $\Lambda([a,b]\times\overline{\omega})$ is a compact subset of $(0,\delta_{0})\times{\mathbb{R}}^{N}$. Let $1\leq q<\infty$. Since $v\in C((0,\delta_{0}),H^{2}({\mathbb{R}}^{N}))$ and $H^{2}({\mathbb{R}}^{N})\hookrightarrow L^{r}({\mathbb{R}}^{N})$ for every $r<\infty$ (because $N\leq 4$), we have $v\in L^{q}(\Lambda((a,b)\times\omega))$; and so (4.27) follows from (4.25) and the change of variable formula. Next, let $\theta\in C^{\infty}_{\mathrm{c}}((0,\delta_{0})\times{\mathbb{R}}^{N})$ such that $\theta(x)\equiv 1$ on $\Lambda([a,b]\times\overline{\omega})$. Thus we may replace $v$ by $\theta v$ in formula (4.25), this does not change the values of $u$ on $(a,b)\times\omega$. Since $\theta v\in H^{2}((0,\delta_{0})\times{\mathbb{R}}^{N})$, we can approximate $\theta v$ in $H^{2}((0,\delta_{0})\times{\mathbb{R}}^{N})$ by a sequence $(w_{n})_{n\geq 1}\subset C^{\infty}_{\mathrm{c}}((0,\delta_{0})\times{\mathbb{R}}^{N})$ supported in a fixed compact of $(0,\delta_{0})\times{\mathbb{R}}^{N}$. Setting $u_{n}=w_{n}\circ\Lambda$, we have

[TABLE]

Next, it follows from (4.25) that

[TABLE]

and

[TABLE]

where the argument of $\psi$ is $y$ and the argument of $w_{n}$ is $\Lambda$.

Similar formulas hold for all first and second space-time derivatives of $u_{n}$, so arguing as in (4.29) we conclude that $u_{n}$ is a Cauchy sequence in $H^{2}((a,b)\times\omega)$, from which (4.26) follows. In addition, the above two formulas imply that

[TABLE]

Since $u_{n}\to u$ in $H^{2}((a,b)\times\omega)$ and $w_{n}\to\theta v$ in $H^{2}((0,\delta_{0})\times{\mathbb{R}}^{N})$, we may pass to the limit in the above equation. Since $\theta v=v$ in $\Lambda((a,b)\times\omega)$, we obtain using (2.2)

[TABLE]

in $L^{2}((a,b)\times\omega)$. This proves (4.28).

Set

[TABLE]

and

[TABLE]

so that $\widetilde{\tau}>0$ by (4.17). We see that $(0,\widetilde{\tau})\times B_{\widetilde{\rho}}\subset\mathcal{T}$ so that $u\in H^{2}((0,\widetilde{\tau})\times B_{\widetilde{\rho}})\cap L^{q}((0,\widetilde{\tau})\times B_{\widetilde{\rho}})$ for all $q<\infty$. In particular, $u\in C([0,\widetilde{\tau}],H^{1}(B_{\widetilde{\rho}}))\cap C^{1}([0,\widetilde{\tau}],L^{2}(B_{\widetilde{\rho}}))$, so that $u(0)\in H^{1}(B_{\widetilde{\rho}})$ and $\partial_{t}u(0)\in L^{2}(B_{\widetilde{\rho}})$ are well defined.

4.5. Choice of a solution of the nonlinear wave equation

We apply Section 1.2 to extend $u$, which is a solution of (1.1) on $\mathcal{T}$, to a solution of (1.1) on a maximal domain of influence that contains $\mathcal{T}$. For this, we consider any pair $(\widetilde{u}_{0},\widetilde{u}_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$ such that $\widetilde{u}_{0}$ and $\widetilde{u}_{1}$ coincide with $u(0)$ and $\partial_{t}u(0)$, respectively, on $B_{\widetilde{\rho}}$. The initial data $(\widetilde{u}_{0},\widetilde{u}_{1})$ give rise to a solution $\widetilde{u}$ of (1.1) defined on the maximal influence domain $\Omega_{\rm max}(\widetilde{u}_{0},\widetilde{u}_{1})$ in the sense of §1.2. We claim that this maximal influence domain contains

[TABLE]

and that $\widetilde{u}$ coincides with $u$ on $\widetilde{\mathcal{T}}$. Indeed, let $(t,x)\in\widetilde{\mathcal{T}}$ and consider the corresponding open backward cone $C(t,x)$. The cone $C(t,x)$ is an influence domain, and It follows easily, using Proposition B.2 and (4.27), that $u$ is a solution of (1.1) in $C(t,x)$ with initial data $(u_{0},u_{1})$, so that $C(t,x)\subset\Omega_{\rm max}(\widetilde{u}_{0},\widetilde{u}_{1})$. Since $(t,x)\in\widetilde{\mathcal{T}}$ is arbitrary, this proves the claim. From now on, we denote by $u$ this solution.

4.6. Blowup on the local hypersurface and end of the proof

We show blowup on the local hypersurface by proving (1.4). For this, we further restrict the size of the hypersurface. Arguing as in the proof of (4.18), we see that

[TABLE]

Thus we see that if $|x_{0}|\leq\frac{\varepsilon_{0}}{4}$, then the open backward cone $C(\tau_{0}+\widetilde{\varphi}(x_{0}),x_{0})$ is a subset of $\widetilde{\mathcal{T}}$.

We fix $\ell<\sigma\leq 1$ and $|x_{0}|\leq\frac{\varepsilon_{0}}{4}$, and we prove (1.4). We use the geometric property that the image by the map $\Lambda$ of a cone of slope $\sigma$ contains at least a small cone (estimate (4.31)), and the lower estimate (3.6) for $v$ on this small cone.

Let $s_{0}\geq 0$ and $y_{0}\in{\mathbb{R}}^{N}$ be given by $\Lambda(\tau_{0}+\widetilde{\varphi}(x_{0}),x_{0})=(s_{0},y_{0})$. We first note that $s_{0}=0$ by (4.12) and (4.19). Moreover, it follows from (4.19), (4.3) and (4.16) that

[TABLE]

Given $0\leq t<\tau_{0}+\widetilde{\varphi}(x_{0})$, we set

[TABLE]

and, given $s>0$ and $\sigma^{\prime}>0$ we set

[TABLE]

We claim that there exist $\sigma^{\prime}>0$ and $\eta>0$ such that

[TABLE]

where

[TABLE]

Assuming (4.31)-(4.32), we conclude the proof of (1.4). Given $(t,x)\in\widetilde{\mathcal{T}}$, it follows from (4.25) and (4.19) that

[TABLE]

so that, using $2\ell xy\leq\ell^{2}x^{2}+y^{2}$,

[TABLE]

Therefore,

[TABLE]

Applying (3.6) and (4.31), we deduce that

[TABLE]

It follows from (4.32), (2.14) and (4.30) that

[TABLE]

Furthermore,

[TABLE]

Next, $H^{1}({\mathbb{R}}^{N})\hookrightarrow L^{\frac{2(N+2)}{N}}({\mathbb{R}}^{N})$, so that $g^{2}\in L^{\frac{N+2}{N}}((0,\delta_{0})\times{\mathbb{R}}^{N}))$; and so by (4.35)

[TABLE]

Estimate (1.4) follows from (4.33)–(4.36).

It remains to prove the claim (4.31)-(4.32). Let $(s^{\prime},y)\in{\mathbb{R}}^{1+N}_{+}$ and $(t^{\prime},x)\in{\mathbb{R}}^{1+N}$ such that $(s^{\prime},y)=\Lambda(t^{\prime},x)$. In particular, $t^{\prime}\leq\tau_{0}+\widetilde{\varphi}(x)$ by (4.20). We prove that

[TABLE]

In the case $\ell=0$, this follows from (4.22) and the inequality $|\widetilde{\varphi}(x)-\widetilde{\varphi}(x_{0})|\leq|x-x_{0}|$ (see (4.3)). In the case $\ell\not=0$, then by (4.19) and (4.23),

[TABLE]

Using the right-hand side inequality in (4.9), and then (4.3), we deduce

[TABLE]

and (4.37) by using again (4.3).

Next we claim that

[TABLE]

Indeed, by (4.19) for $(t^{\prime},x)$ and for $(\tau_{0}+\widetilde{\varphi}(x_{0}),x_{0})$,

[TABLE]

so that

[TABLE]

Estimate (4.38) follows by using the triangle inequality $\sqrt{(a+b)^{2}+c^{2}}\leq\sqrt{a^{2}+c^{2}}+|b|$. Assuming now $(s^{\prime},y)\in L(s,\sigma^{\prime})$ for some $s>0$ and $\sigma^{\prime}>0$, we deduce from (4.38) that

[TABLE]

Estimating $s^{\prime}$ by (4.37), we obtain

[TABLE]

Since $\sigma>\ell$, we see that if $\sigma^{\prime}>0$ and $\delta>0$ are sufficiently small, then

[TABLE]

It now remains to prove that if $s^{\prime}\leq\eta(\tau_{0}+\widetilde{\varphi}(x_{0})-t)$ for some sufficiently small $\eta>0$, then $t^{\prime}\geq t$. By (4.24), and then (4.3), we deduce

[TABLE]

Using (4.39) we obtain

[TABLE]

which proves the claim for $\eta=(1-\sigma+\delta)(\frac{1-\ell}{1+\ell})^{\frac{1}{2}}$.

Finally, we prove that the hypersurface $\{(t,x)\in{\mathbb{R}}^{1+N}_{+};\,|x_{0}|<\frac{\varepsilon_{0}}{4},t=\tau_{0}+\widetilde{\varphi}(x_{0})\}$ is contained in the upper boundary of the maximal influence domain $\Omega_{\mathrm{max}}$ of the solution $u$. Indeed, otherwise there would exist $|x_{0}|<\frac{\varepsilon_{0}}{4}$ and $t>\tau_{0}+\widetilde{\varphi}(x_{0})$ such that $C(t,x_{0})\subset\Omega_{\mathrm{max}}$ with the notation (1.5). In particular,

[TABLE]

This is absurd, since by (1.4), given $\ell<\sigma\leq 1$, there exist a sequence $t_{n}\uparrow\tau_{0}+\widetilde{\varphi}(x_{0})$ and $\delta>0$ such that

[TABLE]

This completes the proof of the theorem, where $\tau_{0}$ and $\varepsilon_{0}$ are given by (4.16), and $\varepsilon=\min\{\frac{\varepsilon_{0}}{4},r\}$ with $r$ defined in Section 1.1 (recall that $\varphi=\widetilde{\varphi}$ on $\{|x|<r\}$).

Appendix A The wave equation (2.2)

Let $\psi\in{\mathcal{C}}^{2}({\mathbb{R}}^{N})\cap W^{2,\infty}({\mathbb{R}}^{N})$ satisfy $\|\nabla\psi\|_{L^{\infty}}<1$. It follows in particular that $(1-|\nabla\psi|^{2})^{-1}\in C^{1}({\mathbb{R}}^{N})\cap W^{1,\infty}({\mathbb{R}}^{N})$.

A.1. The associated semigroup

Let $X$ be the Hilbert space $H^{1}\times L^{2}$, equipped with the (equivalent) scalar product

[TABLE]

and consider the linear operator $\mathcal{A}$ on $X$ defined by

[TABLE]

with domain $D(\mathcal{A})=H^{2}\times H^{1}$. We compute

[TABLE]

which proves that $\mathcal{A}$ is dissipative in $X$. Moreover, for any $(c,d)\in X$, there exist $(a,b)\in D(\mathcal{A})$ such that $(a,b)-\mathcal{A}(a,b)=(c,d)$. Indeed, this system reduces to

[TABLE]

It is easy to solve the second equation by the Lax-Milgram theorem, and we obtain a solution $a\in H^{1}({\mathbb{R}}^{N})$. Since, by the equation, $\Delta a\in L^{2}({\mathbb{R}}^{N})$, we see that $a\in H^{2}({\mathbb{R}}^{N})$. The first equation then yields $b\in H^{1}({\mathbb{R}}^{N})$. In particular $\mathcal{A}$ is maximal dissipative, hence is the generator of a $C_{0}$ semigroup of contractions $(e^{t\mathcal{A}})_{t\geq 0}$ on $X$. (See e.g. Chapter 1, Theorem 4.3, p. 14 in [27].)

A.2. The nonlinear equation

Using the notation $U=\binom{v}{\partial_{s}v}$, we rewrite equation (2.2) as

[TABLE]

where

[TABLE]

A.3. Regularity

Suppose $T>0$ and $U\in L^{\infty}((0,T),D(\mathcal{A}))\cap W^{1,\infty}((0,T),X)$ is such that $\mathcal{F}(U)\in L^{\infty}((0,T),X)$ and $U$ satisfies equation (A.1) for a.a. $0<t<T$. If $\mathcal{F}(U)\in C((0,T),D(\mathcal{A}))$, then $U\in C((0,T),D(\mathcal{A}))\cap C^{1}((0,T),X)$. Indeed, $U$ is weakly continuous $(0,T)\to D(\mathcal{A})$. In particular, $U(t)\in D(\mathcal{A})$ for all $0<t<T$ and the result follows easily, see e.g. Chapter 4, Corollary 2.6, p. 108 in [27].

A.4. The case of equation (3.7)

Equation (3.7) is equation (A.1), where $f$ is replaced by $f_{n}$ in (A.2). Since $f_{n}(0)=0$ and $f_{n}$ is globally Lipschitz ${\mathbb{R}}\to{\mathbb{R}}$, we see that the map $u\mapsto f_{n}(u)$ is globally Lipschitz $L^{2}({\mathbb{R}}^{N})\to L^{2}({\mathbb{R}}^{N})$. In particular, $\mathcal{F}:X\to X$ is globally Lipschitz, and the existence and uniqueness of a global, mild solution $U\in C([0,\infty),X)$ of (A.1) with the initial condition $U(0)=U_{0}\in X$ is a direct consequence of standard semigroup theory. (See e.g. Chapter 6, Theorem 1.2, p. 184 in [27].) Moreover, since $f_{n}$ is globally Lipschitz and $C^{1}$, it follows easily that the map $u\mapsto f_{n}(u)$ is continuous $H^{1}({\mathbb{R}}^{N})\to H^{1}({\mathbb{R}}^{N})$. Therefore $\mathcal{F}$ is continuous $X\to D(\mathcal{A})$, so that $\mathcal{F}(U)\in C([0,\infty),D(\mathcal{A}))$. It follows, again by the semigroup theory, that if the initial value is in $D(\mathcal{A})$, then $U\in C([0,\infty),D(\mathcal{A}))\cap C^{1}([0,\infty),X)$ is a solution of (A.1). (See e.g. Chapter 4, Corollary 2.6, p. 108 in [27].)

Appendix B Uniqueness on light cones

We state and prove a uniqueness property for solutions of the nonlinear wave equation on light cones (Proposition B.2), for which we could not find a reference. We first recall in the following remark the relevant results concerning the local well-posedness of the Cauchy problem.

Remark B.1** (Local well-posedness).**

Let $N\geq 1$, let $p$ such that $1<p\leq\frac{N+2}{N-2}$ ($1<p<\infty$ if $N=1,2$) and let $(u_{0},u_{1})\in H^{1}({\mathbb{R}}^{N})\times L^{2}({\mathbb{R}}^{N})$. We summarize some results on the existence of $T>0$ and a local solution

[TABLE]

of the wave equation

[TABLE]

We also discuss the property

[TABLE]

in the case $N\geq 3$.

1. (i)

Case $N=1,2$. There exist $T>0$ and a unique solution $u$ of (B.2) in the class (B.1). See e.g. [5, Theorem 6.2.2]. 2. (ii)

Case $N\geq 3$, $p<\frac{N+2}{N-2}$. There exist $T>0$ and a unique solution $u$ of (B.2) in the class (B.1), and this solution satisfies (B.3) by possibly choosing $T$ smaller. Indeed, existence follows from [12, Proposition 2.3] and uniqueness from [11, Proposition 3.1]. Moreover, applying Lemma 3.3 in [11] with $\rho=\frac{N}{2(N-1)}$, $r=\frac{2(N^{2}-1)}{N^{2}-2N+3}$ and $q=\frac{2(N+1)}{N-1}$, we see that $u\in L^{q}((0,T),\dot{B}^{\rho}_{r,2}({\mathbb{R}}^{N}))$, hence (B.3) by Sobolev’s embedding. 3. (iii)

Case $N=3$, $p=5$. There exist $T>0$ and a solution $u$ of (B.2) in the class (B.1)-(B.3). See e.g. [15, Theorem 2.7]. Moreover, solutions of (B.2) in the class (B.1)-(B.3) are unique. This last property is not explicitly stated in [15], but it easily follows from the proof. (It also follows from Proposition B.2.) 4. (iv)

Case $N\geq 4$, $p=\frac{N+2}{N-2}$. There exist $T>0$ and a unique solution $u$ of (B.2) in the class (B.1), and this solution satisfies property (B.3) by possibly choosing $T$ smaller. Indeed, existence is established in [10] (see also [15, Theorem 2.7] for the case $N=4,5$ and [2, Theorem 3.3] for the case $N\geq 6$). Uniqueness is proved in [28, Theorem 2] for $N=4$, in [28, Theorem 3] for $N=5$ and in [2, Theorem 3.4] for $N\geq 6$. Property (B.3) follows from [15, Theorem 2.7] in the case $N=4,5$. In the case $N\geq 6$, it follows from [2, Theorem 3.3] that $u\in L^{q}((0,T),\dot{B}^{\rho}_{r,2}({\mathbb{R}}^{N}))$ with $\rho=\frac{N}{2(N-1)}$, $r=\frac{2(N^{2}-1)}{N^{2}-2N+3}$ and $q=\frac{2(N+1)}{N-1}$, hence (B.3) by Sobolev’s embedding.

Proposition B.2** (Uniqueness on light cones).**

Let $N\geq 1$ and let $p$ satisfy $1<p\leq\frac{N+2}{N-2}$ ($1<p<\infty$ if $N=1,2$). Let $R>0$, $0<\tau<R$, and let $B_{R}$ be the open ball of center [math] and radius $R$ in ${\mathbb{R}}^{N}$. Let

[TABLE]

be two solutions of the wave equation $\partial_{tt}u=\Delta u+|u|^{p-1}u$ in $H^{-1}(B_{R}))$ for $\leq t\leq\tau$. If $N\geq 3$ and $p>\frac{N}{N-2}$, suppose in addition that $u,v\in L^{\frac{2(N+1)}{N-2}}((0,T)\times B_{R})$. If $u(0)=v(0)$ and $\partial_{t}u(0)=\partial_{t}v(0)$, then $u=v$ on $\{(t,x)\in(0,\tau)\times B_{R};\,|x|<R-t\}$.

The proof of Proposition B.2 relies on the following local estimates.

Lemma B.3**.**

Let $R>0$, $0<\tau<R$, $h\in C([0,\tau],L^{q}(B_{R}))$ for some $q\geq 1$, $q\geq\frac{2N}{N+2}$ (so that $h\in C([0,\tau],H^{-1}(B_{R}))$), and let

[TABLE]

satisfy $\partial_{tt}z=\Delta z+h$ in $H^{-2}(B_{R})$ for all $0\leq t\leq\tau$ and if $z(0)=\partial_{t}z(0)=0$. If $h_{|E(0,R,\tau)}\in L^{2}(E(0,R,\tau))$ with the notation (1.6), then $z(t)\in H^{1}(B_{R-t)}$ for all $0<t<\tau$, and

[TABLE]

for all $0<t<\tau$. If $N\geq 2$ and $h\in L^{\frac{2(N+1)}{N+3}}(E(0,R,\tau))$, then $z_{|E(0,R,\tau)}\in L^{\frac{2(N+1)}{N-1}}(E(0,R,\tau))$ and

[TABLE]

In (B.4) and (B.5), the constant $C$ independent of $h$, $R$, $\tau$ and $t$.

Proof.

We define $\widetilde{h}\in C([0,\tau],L^{q}({\mathbb{R}}^{N}))$ by

[TABLE]

We let $\widetilde{z}\in C([0,\tau],L^{2}({\mathbb{R}}^{N}))\cap C^{1}([0,\tau],H^{-1}({\mathbb{R}}^{N}))\cap C^{2}([0,\tau],H^{-2}({\mathbb{R}}^{N}))$ be the solution of the wave equation $\partial_{tt}\widetilde{z}-\Delta\widetilde{z}=\widetilde{h}$ on ${\mathbb{R}}^{N}$ with the initial conditions $\widetilde{z}(0)=\partial_{t}\widetilde{z}(0)=0$. Note that, given any $0<t\leq\tau<R$ and $1\leq r\leq\infty$, $\|\widetilde{h}\|_{L^{r}(E(0,R,t))}=\|h\|_{L^{r}(E(0,R,t))}$. Therefore, estimate (B.4) with $z$ replaced by $\widetilde{z}$ follows from the standard energy inequality for $\widetilde{z}$; and (B.5) with $z$ replaced by $\widetilde{z}$ follows from the Strichartz estimates (see [13, Corollary 1.3]).

To conclude the proof, we show that $z$ and $\widetilde{z}$ coincide on $E(0,R,\tau)$. We let $w(t)=(z(t)-\widetilde{z}(t))_{|B_{R}}$ for all $0\leq t\leq\tau$, so that

[TABLE]

satisfies $\partial_{tt}w=\Delta w$ in $H^{-2}(B_{R})$ for all $0\leq t\leq\tau$ and $w(0)=\partial_{t}w(0)=0$. Thus we need to show that $w=0$ a.e. on $E(0,R,\tau)$. Let $\rho\in C^{\infty}_{\mathrm{c}}({\mathbb{R}}^{N})$, $\rho\geq 0$, be radially symmetric, supported in $B_{1}$, and satisfy $\int\rho=1$. Given $\varepsilon>0$, let $\rho_{\varepsilon}(x)=\varepsilon^{-N}\rho(\frac{x}{\varepsilon})$. Let $0<\eta<R$ and $0<\varepsilon<\eta/2$. Since $\rho_{\varepsilon}$ is supported in $B_{\varepsilon}$, it follows that $\rho_{\varepsilon}\star w$ is well defined in $B_{R-\eta}$, and we set $w_{\varepsilon}=(\rho_{\varepsilon}\star w)_{|B_{R-\eta}}$. We claim that

[TABLE]

By finite speed of propagation, it follows that $w_{\varepsilon}$ identically vanishes on $E(0,R-\eta,\tau)$. Letting $\varepsilon\to 0$, we deduce that $w$ vanishes a.e. on $E(0,R-\eta,\tau)$; and letting $\eta\to 0$ we see that $w$ vanishes a.e. on $E(0,R,\tau)$. It remains to prove the claims (B.7)-(B.9). Given $m\in{\mathbb{N}}$ and $\theta\in H^{-m}(B_{R}))$, recall that $\rho_{\varepsilon}\star\theta\in H^{-m}(B_{R-\eta}))$ is given by

[TABLE]

for all $\varphi\in C^{\infty}_{\mathrm{c}}(B_{R-\eta})$. It is well known that $\rho_{\varepsilon}\star\theta\in C^{\infty}(\overline{B_{R-\eta}}))$, $D^{\alpha}(\rho_{\varepsilon}\star\theta)=\rho_{\varepsilon}\star(D^{\alpha}\theta)$ for all $\alpha\in{\mathbb{N}}^{N}$, and $\|\rho_{\varepsilon}\star\theta\|_{C(\overline{B_{R-\eta}})}\lesssim\|\theta\|_{H^{-m}(B_{R})}$. On the other hand, it follows from (B.6) that $D^{\alpha}\partial_{t}^{\beta}w\in C([0,\tau],H^{-2}(B_{R}))$ for all $\alpha\in{\mathbb{N}}^{N}$ and $\beta\in{\mathbb{N}}$ such that $|\alpha|+\beta\leq 2$. Thus we see that $D^{\alpha}\partial_{t}^{\beta}w_{\varepsilon}\in C([0,\tau]\times\overline{B_{R-\eta}})$ and that $D^{\alpha}\partial_{t}^{\beta}w_{\varepsilon}=\rho_{\varepsilon}\star D^{\alpha}\partial_{t}^{\beta}w$. Properties (B.7)-(B.9) easily follow. ∎

Proof of Proposition B.2.

We need only prove the result for $\tau$ small, the general case follows by iteration.

The case $p\leq\frac{N}{N-2}$ (any $1<p<\infty$ if $N=1,2$). We note that $r=2p$ satisfies $2<r\leq\frac{2N}{N-2}$ ($2<r<\infty$ if $N=1,2$), so that by (B.4) and Sobolev’s embedding

[TABLE]

Since $\|\,|u|^{p-1}u-|v|^{p-1}v\|_{L^{2}}\leq C(\|u\|_{L^{r}}+\|v\|_{L^{r}})^{p-1}\|u-v\|_{L^{r}}$ and $u$ and $v$ are bounded in $H^{1}$, hence in $L^{r}$, the result follows by Gronwall’s inequality.

The case $N\geq 3$ and $p=\frac{N+2}{N-2}$. We note that, since $p\leq\frac{N+2}{N-2}$,

[TABLE]

By Hölder’s inequality, it follows that

[TABLE]

where all the integrals are on $E(0,R,\tau)$, with the notation (1.6). Applying the Strichartz inequality (B.5), we deduce that

[TABLE]

where all the integrals are on $E(0,R,\tau)$. Since

[TABLE]

the conclusion follows by choosing $\tau$ sufficiently small. ∎