Exterior Energy Bounds for the Critical Wave Equation Close to the Ground State

Abstract

By definition, the exterior asymptotic energy of a solution to a wave equation on ${\mathbb {R}}^{1+N}$ is the sum of the limits as $t\rightarrow \pm \infty$ of the energy in the the exterior $\{|x|>|t|\}$ of the wave cone. In our previous work Duyckaerts et al. (J Eur Math Soc 14(5):1389–1454, 2012), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension N is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential

$$\begin{aligned} \partial _t^2u+L_Wu=0,\quad L_W:=-\Delta -\frac{N+2}{N-2}W^{\frac{4}{N-2}} \end{aligned}$$

(*)

obtained by linearizing the energy critical wave equation at the ground-state solution W, still in odd space dimension. This equation admits nonzero solutions of the form $A+tB$, where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of ($*$) is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of W, and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions 3 and 5.

Appendix A: Lorentz Transformation

This appendix concerns the effect of the Lorentz transformations on solutions of (1.1). If u is a $C^2$ classical solution of (1.1), then by direct computation, $u_{\varvec{\ell }}(t,x)$ (defined by (1.4) is also a $C^2$ classical solution of (1.1) on its domain of definition. The Lorentz tranform of a general finite energy solution of (1.1) (as defined in Definition 5.3 above) is more difficult to understand. If u is global, the formula (1.4) makes sense, and one can prove that $u_{\varvec{\ell }}$ has indeed finite energy and is a solution of (1.1) in the sense of Definition 5.3 (see e.g. [12, Lemma 6.1]).

If u is not globally defined, the formula (1.4) does not make sense anymore. In this section we prove however that using the Definition 5.6 of solutions of (1.1) outside wave cones, we can define the Lorentz transformation of a class of nonglobal solutions, that include a neighborhood of any global solution.

If $\varvec{\ell }\in {\mathbb {R}}^N$ with $|\varvec{\ell }|<1$, we denote by

$$\begin{aligned} c_{\varvec{\ell }}:=\sqrt{\frac{1+|\varvec{\ell }|}{1-|\varvec{\ell }|}}>1. \end{aligned}$$

Let $(t,x)\in {\mathbb {R}}^N$, and (s, y) given by the change of variable of the Lorentz transformation:

$$\begin{aligned} (s,y)=\left( \frac{t-\varvec{\ell }\cdot x}{\sqrt{1-\ell ^2}},\left( -\frac{t}{\sqrt{1-\ell ^2}}+\frac{1}{\ell ^2} \left( \frac{1}{\sqrt{1-\ell ^2}}-1\right) \varvec{\ell }\cdot x\right) \varvec{\ell }+x\right) . \end{aligned}$$

Then

$$\begin{aligned} |x|^2-t^2=|y|^2-s^2 \end{aligned}$$

and

$$\begin{aligned} |s|+|y|\le c_{\varvec{\ell }}(|t|+|x|),\quad |t|+|x|\le c_{\varvec{\ell }}(|s|+|y|). \end{aligned}$$

This can be checked easily, assuming for example that $\varvec{\ell }=(\ell ,0,\ldots ,0)$, so that

$$\begin{aligned} (s,y)=\left( \frac{t-\ell x_1}{\sqrt{1-\ell ^2}},\frac{x_1-t\ell }{\sqrt{1-\ell ^2}},x_2,\ldots ,x_N \right) . \end{aligned}$$

(A.1)

Lemma A.1

Let $\eta _0\in (0,1)$. There exists $T>0$ with the following property. Let $\tau \ge T$, u be a scattering solution of (1.1) in $\{|x|>|t|-\tau \}$ with initial data $(u_0,u_1)\in {\mathcal {H}}$ at $t=0$, and $\varvec{\ell }\in {\mathbb {R}}^N$ with $|\varvec{\ell }|\le \eta _0$. Then the formula (1.4) makes sense for $t\in [-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ]$ and defines a solution of (1.1) on $[-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ]\times {\mathbb {R}}^N$. Furthermore,

$$\begin{aligned} E(\vec {u}_{\varvec{\ell }}(0))&=\frac{E(u_0,u_1)}{\sqrt{1-|\varvec{\ell }|^2}} -\frac{1}{\sqrt{1-|\varvec{\ell }|^2}} \varvec{\ell }\cdot P(u_0,u_1) \end{aligned}$$

(A.2)

$$\begin{aligned} P(\vec {u}_{\varvec{\ell }}(0))&=P(u_0,u_1)+\frac{\varvec{\ell }\cdot P(u_0,u_1)}{|\varvec{\ell }|^2}\left( \frac{1}{\sqrt{1-|\varvec{\ell }|^2}}-1\right) \varvec{\ell }-\frac{E(u_0,u_1)}{\sqrt{1-|\varvec{\ell }|^2}} \varvec{\ell }. \end{aligned}$$

(A.3)

Lemma A.2

Let $\eta _0$, $\tau$ and u be as in Lemma A.1. There exist constants $\varepsilon _0>0$ and $C>0$ (depending on u, $\tau$ and $\eta _0$) such that if $(v_0,v_1)\in {\mathcal {H}}$ and

$$\begin{aligned} \Vert (u_0,u_1)-(v_0,v_1)\Vert _{{\mathcal {H}}}<\varepsilon _0, \end{aligned}$$

then the solution v of (1.1) in $\{|x|>|t|-\tau \}$ with initial data $(v_0,v_1)$ at $t=0$ is scattering, and, if $|\varvec{\ell }|\le \eta _0$,

$$\begin{aligned} \left\| \vec {u}_{\varvec{\ell }}(0)-\vec {v}_{\varvec{\ell }}(0)\right\| _{{\mathcal {H}}}\le C\left\| (u_0,u_1)-(v_0,v_1)\right\| _{{\mathcal {H}}}. \end{aligned}$$

Remark A.3

Let u be a global solution of (1.1). Then by [14], we can see that for all $A\in {\mathbb {R}}$,

$$\begin{aligned} u1\!\!1_{|x|\ge |t|+A}\in L^{\frac{N+2}{N-2}}\left( {\mathbb {R}},L^{\frac{2(N+2)}{N-2}}({\mathbb {R}}^N)\right) . \end{aligned}$$

Thus Lemma A.1 applies and one can define the Lorentz transform $u_{\varvec{\ell }}$ (which is global) of u for any parameter $\varvec{\ell }$, with $|\varvec{\ell }|<1$. Furthermore by Lemma A.2, for all $\eta _0$, there exists $\varepsilon _0$ such that if $\Vert (u_0,u_1)-(v_0,v_1)\Vert _{{\mathcal {H}}}<\varepsilon _0$ and $|\varvec{\ell }|\le \eta _0$, then one can define the Lorentz transform $v_{\varvec{\ell }}$ of the solution v of (1.1) with initial data $(v_0,v_1)$.

1.1 A.1 Lorentz transform of a solution

In this subsection we prove the first part of Lemma A.1, i.e. the fact that $u_{\varvec{\ell }}(t)$ is well-defined for $t\in [-c_{\varvec{\ell }}\tau ,c_{\varvec{\ell }}\tau ]$. We assume without loss of generality

$$\begin{aligned} \varvec{\ell }=(\ell ,0,\ldots ,0). \end{aligned}$$

We recall from [25, Lemma 2.2 and Remark 2.3] the following claim:

Claim A.4

Let $\eta _0\in (0,1)$, $h\in L^1({\mathbb {R}},L^2({\mathbb {R}}^N))$, $(w_0,w_1)\in {\dot{H}}^1\times L^2$, $\varvec{\ell }\in {\mathbb {R}}^N$ with $|\varvec{\ell }|\le \eta _0$ and

$$\begin{aligned} w(t)=\cos (t\sqrt{-\Delta })w_0+\frac{\sin (t\sqrt{-\Delta })}{\sqrt{-\Delta }}w_1 +\int _0^t\frac{\sin \left( (t-s)\sqrt{-\Delta } \right) }{\sqrt{-\Delta }}h(s)\,ds,\quad t\in {\mathbb {R}}.\nonumber \\ \end{aligned}$$

(A.4)

Then $(w_{\varvec{\ell }},\partial _tw_{\varvec{\ell }})\in C^0\left( {\mathbb {R}},{\dot{H}}^1\times L^2\right)$ and there is a constant $C_{\eta _0}$ (depending only on $\eta _0$) such that

$$\begin{aligned} \sup _{t} \left\| (w_{\ell }(t),\partial _tw_{\ell }(t)\right\| _{{\dot{H}}^1\times L^2}\le C_{\eta _0}\left( \Vert (w_0,w_1)\Vert _{{\dot{H}}^1\times L^2}+\Vert h\Vert _{L^1({\mathbb {R}},L^2)} \right) . \end{aligned}$$

Step 1

(Smooth compactly-supported initial data) We first assume $(u_0,u_1)\in \left( C_0^{\infty }({\mathbb {R}}^N)\right) ^2$, and denote by R a positive number such that $(u_0,u_1)(y)=0$ for $| y|\ge R$. We denote by ${\mathcal {E}}$ the exterior of the wave cone:

$$\begin{aligned} {\mathcal {E}}:=\left\{ (s,y)\in {\mathbb {R}}\times {\mathbb {R}}^N\;|\;|y|>|s|-\tau \right\} , \end{aligned}$$

and ${\mathcal {E}}_{\varvec{\ell }}$ its image:

$$\begin{aligned} {\mathcal {E}}_{\varvec{\ell }}&:=\left\{ \left( \frac{s+\ell y_1}{\sqrt{1-\ell ^2}},\frac{y_1+s\ell }{\sqrt{1-\ell ^2}},y_2,\ldots ,y_N \right) ,\; (s,y)\in {\mathcal {E}}\right\} \\&= \bigg \{(t,x)\in {\mathbb {R}}\times {\mathbb {R}}^N\;\Big |\;\left( \frac{t-\ell x_1}{\sqrt{1-\ell ^2}},\frac{x_1-t\ell }{\sqrt{1-\ell ^2}},x_2,\ldots ,x_N \bigg )\in {\mathcal {E}}\right\} . \end{aligned}$$

One can prove

$$\begin{aligned} u\in C^0({\mathcal {E}}). \end{aligned}$$

(A.5)

Indeed, since the nonlinearity F is $C^2$, we have that for all $(s_0,y_0)\in {\mathcal {E}}$, there exists a neighborhood $J\times \omega$ of $(s_0,y_0)$ in ${\mathcal {E}}$ such that

$$\begin{aligned} \vec {u}\in C^0\left( J,(H^{3}\times H^2)(\omega ) \right) , \end{aligned}$$

and (A.5) follows from Sobolev embedding (recall that $N\le 5$). By (A.5) and the definition of $u_{\varvec{\ell }}$,

$$\begin{aligned} u_{\varvec{\ell }}\in C^0({\mathcal {E}}_{\varvec{\ell }}). \end{aligned}$$

(A.6)

We next prove that if t satisfies $|t|\le c_{\varvec{\ell }}^{-1}\tau$ and $x\in {\mathbb {R}}^N$, then $(t,x)\in {\mathcal {E}}_{\varvec{\ell }}$. Indeed, letting (s, y) be as in (A.1), we must prove that $(s,y)\in {\mathcal {E}}$. We have

$$\begin{aligned} |y|-|s|=\frac{|y|^2-s^2}{|y|+|s|}=(|x|-|t|)\frac{|x|+|t|}{|y|+|s|}\ge -|t|\frac{|x|+|t|}{|y|+|s|}\ge -c_{\varvec{\ell }}^{-1}|\tau |\frac{|x|+|t|}{|y|+|s|}. \end{aligned}$$

Since $\frac{|x|+|t|}{|y|+|s|}\le c_{\varvec{\ell }}$, we deduce $(s,y)\in {\mathcal {E}}$, i.e. $(t,x)\in {\mathcal {E}}_{\varvec{\ell }}$. Using that $1\!\!1_{|y|\ge |s|-|\tau |}u\in L^{\frac{N+2}{N-2}}({\mathbb {R}},L^{\frac{2(N+2)}{N-2}})$, we obtain by Claim A.4

$$\begin{aligned} \vec {u}_{\varvec{\ell }}\in C^0\left( [-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ],{\mathcal {H}}\right) . \end{aligned}$$

(A.7)

Next, we prove

$$\begin{aligned} |t|\le c_{\varvec{\ell }}^{-1}\tau ,\; |x|\ge |t|+R\,c_{\varvec{\ell }}\Longrightarrow u_{\varvec{\ell }}(t,x)=0 \end{aligned}$$

Indeed, the left-hand side of this implication implies

$$\begin{aligned} |y|-|s|\ge \frac{|x|+|t|}{|y|+|s|} c_{\varvec{\ell }}\,R\ge R, \end{aligned}$$

and thus $u_{\varvec{\ell }}(t,x)=u(s,y)=0$.

Since $u_{\varvec{\ell }}$ is compactly supported in the space variable and continuous on $\left[ -\frac{\tau }{c_{\varvec{\ell }}},\frac{\tau }{c_{\varvec{\ell }}}\right] \times {\mathbb {R}}^N$, we deduce

$$\begin{aligned} u_{\varvec{\ell }}\in L^{\frac{N+2}{N-2}}\left( [-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ],L^{\frac{2(N+2)}{N-2}} \right) . \end{aligned}$$

(A.8)

Finally it is easy to see, using that u satisfies (1.1) in the distributional sense on ${\mathcal {E}}$, that $u_{\varvec{\ell }}$ satisfies (1.1) in the distributional sense on ${\mathcal {E}}_{\varvec{\ell }}$. By Remark 5.4, $u_{\varvec{\ell }}$ is a solution of (1.1) on the interval $[-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ]$. Notice for further use that by a simple change of variables,

$$\begin{aligned} \Vert u_{\varvec{\ell }}\Vert _{L^{\frac{2(N+1)}{N-2}}\left( [-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ]\times {\mathbb {R}}^N\right) }\lesssim \Vert u\Vert _{L^{\frac{2(N+1)}{N-2}}({\mathcal {E}})}, \end{aligned}$$

(A.9)

where the implicit constant depends only on $\eta _0$.

Step 2

We no longer assume $(u_0,u_1)\in \left( C_0^{\infty }({\mathbb {R}}^N) \right) ^2$, and prove again that the Lorentz transform of u is a solution of (1.1) on $\left[ -c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau \right]$. Let $\left\{ \left( u_{0}^k,u_1^k \right) \right\} _k$ be a sequence in $\left( C_0^{\infty }({\mathbb {R}}^N) \right) ^2$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\left\| \left( u_0^k,u_1^k)-(u_0,u_1) \right) \right\| _{{\mathcal {H}}}=0. \end{aligned}$$

Let u be the solution of (1.1) on ${\mathcal {E}}=\{|x|> |t|-\tau \}$ with initial data $(u_0,u_1)$ at $t=0$. By Definition 5.6, this is the restriction to ${\mathcal {E}}$ of the solution of

$$\begin{aligned} \partial _t^2u-\Delta u=|u|^{\frac{4}{N-2}}u1\!\!1_{{\mathcal {E}}}, \end{aligned}$$

with the same initial data (that we will also denote by u). We let $u^k$ the solution of the same equation with initial data $\left( u_0^k,u_1^k \right)$. By the above computations, the value of $u_{\varvec{\ell }}(t)$ (respectively $u^k_{\varvec{\ell }}(t)$) for $|t|\le c_{\varvec{\ell }}^{-1}\tau$ depends only on the value of u (respectively $u^k$) on ${\mathcal {E}}$. By long-time perturbation theory, we obtain that for large k

$$\begin{aligned} \left\| u^k\right\| _{L^{\frac{2(N+1)}{N-2}}({\mathbb {R}}\times {\mathbb {R}}^N)}\le 2\Vert u\Vert _{L^{\frac{2(N+1)}{N-2}}({\mathbb {R}}\times {\mathbb {R}}^N)}\lesssim \left\| 1\!\!1_{{\mathcal {E}}} u\right\| _{L^{\frac{N+2}{N-2}}\big ({\mathbb {R}},L^{\frac{2(N+2)}{N-2}}\big )}, \end{aligned}$$

(A.10)

and

$$\begin{aligned} \lim _{k\rightarrow \infty } \left\| u-u^k\right\| _{L^{\frac{N+2}{N-2}}\left( {\mathbb {R}},L^{\frac{2(N+2)}{N-2}}\right) }=0. \end{aligned}$$

(A.11)

By the preceding step, $u^k_{\varvec{\ell }}$ is a solution of (1.1) on $\left[ -c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau \right] \times {\mathbb {R}}^N$. Since (A.11) implies

$$\begin{aligned} \lim _{k\rightarrow \infty }\left\| \left( F(u^k)-F(u) \right) 1\!\!1_{{\mathcal {E}}}\right\| _{L^1({\mathbb {R}},L^2)}=0, \end{aligned}$$

we deduce from Claim A.4

$$\begin{aligned} \sup _{t\in [-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ]} \left\| \vec {u}_{\varvec{\ell }}(t)-\vec {u}^k_{\varvec{\ell }}(t)\right\| _{{\mathcal {H}}}\underset{k\rightarrow \infty }{\longrightarrow } 0. \end{aligned}$$

Since by (A.10),

$$\begin{aligned} \left\| u^k_{\varvec{\ell }}\right\| _{L^{\frac{2(N+1)}{N-2}}\left( [-c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau ]\times {\mathbb {R}}^N\right) } \end{aligned}$$

is uniformly bounded (see (A.9) in the preceding step), we deduce by Remark 5.5 that $u_{\varvec{\ell }}$ is a solution of (1.1) on $\left[ -c_{\varvec{\ell }}^{-1}\tau ,c_{\varvec{\ell }}^{-1}\tau \right]$.

1.2 A.2 Perturbation

In this subsection we prove Lemma A.2. We use the notations of the previous subsection. Adapting the standard long-time perturbation theory to the exterior of wave cones, we obtain that there exists $\varepsilon _0$ such that if

$$\begin{aligned} \left\| (v_0,v_1)-(u_0,u_1)\right\| _{{\mathcal {H}}}\le \varepsilon _0, \end{aligned}$$

then the solution v of

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t^2v-\Delta v=F(v)1\!\!1_{{\mathcal {E}}}\\&\vec {v}_{\restriction t=0}=(v_0,v_1) \end{aligned} \right. \end{aligned}$$

scatters and satisfies

$$\begin{aligned} \sup _{t\in {\mathbb {R}}}\left\| \vec {u}(t)-\vec {v}(t)\right\| _{{\mathcal {H}}}+\left\| u-v\right\| _{L^{\frac{N+2}{N-2}}\left( {\mathbb {R}},L^{\frac{2(N+2)}{N-2}}({\mathbb {R}}^N)\right) }\lesssim C\left\| (u_0,u_1)-(v_0,v_1)\right\| _{{\mathcal {H}}}. \end{aligned}$$

As a consequence

$$\begin{aligned} \left\| \left( |v|^{\frac{4}{N-2}}v-|u|^{\frac{4}{N-2}}u\right) 1\!\!1_{{\mathcal {E}}} \right\| _{L^1({\mathbb {R}},L^2)}\lesssim \left\| 1\!\!1_{{\mathcal {E}}}u\right\| _{L^{\frac{N+2}{N-2}}({\mathbb {R}},L^{\frac{2(N+2)}{N-2}})}^{\frac{4}{N-2}}\left\| (u_0,u_1)-(v_0,v_1)\right\| _{{\mathcal {H}}}, \end{aligned}$$

and the conclusion of the Lemma follows from Claim A.4.

1.3 A.3 Energy and momentum

It remains to prove the assertion on the energy and the momentum. This is classical (see e.g. [26]). We give a proof for the sake of completeness. We will assume

$$\begin{aligned} \varvec{\ell }=(\ell ,0,\ldots ,0) \end{aligned}$$

to simplify notations. Let $\zeta \in {\mathbb {R}}$ such that

$$\begin{aligned} \sinh \zeta =\frac{-\ell }{\sqrt{1-\ell ^2}},\quad \cosh \zeta =\frac{1}{\sqrt{1-\ell ^2}}. \end{aligned}$$

As a consequence,

$$\begin{aligned} u_{\varvec{\ell }}(t,x)=u(t\cosh \zeta +x_1\sinh \zeta ,x_1\cosh \zeta +t\sinh \zeta ,x_2,\ldots ,x_N). \end{aligned}$$

(A.12)

Let ${\mathcal {L}}_{\zeta }u(t,x)$ be the right-hand side of (A.12). Formally,

$$\begin{aligned} {\mathcal {L}}_{\zeta +\xi }={\mathcal {L}}_{\zeta }\circ {\mathcal {L}}_{\xi }, \end{aligned}$$

(A.13)

and, by direct computation,

$$\begin{aligned} \frac{d}{d\zeta }E\left( \overrightarrow{{\mathcal {L}}_{\zeta } u}(0)\right) _{\restriction \zeta =0}&=P_1\Big ((u_0,u_1)\Big ) \end{aligned}$$

(A.14)

$$\begin{aligned} \frac{d}{d\zeta }P_1\left( \overrightarrow{{\mathcal {L}}_{\zeta } u}(0)\right) _{\restriction \zeta =0}&=E\Big ((u_0,u_1)\Big ) \end{aligned}$$

(A.15)

$$\begin{aligned} \frac{d}{d\zeta }P_j\left( \overrightarrow{{\mathcal {L}}_{\zeta } u}(0)\right) _{\restriction \zeta =0}&=0,\quad j\in \llbracket 2,N\rrbracket . \end{aligned}$$

(A.16)

where

$$\begin{aligned} P_ju(t)=\int \partial _tu(t,x)\partial _{x_j}u(t,x)\,dx. \end{aligned}$$

Combining (A.13)...(A.16), we deduce

$$\begin{aligned} E\left( \overrightarrow{{\mathcal {L}}_{\zeta }u}(0) \right)&=\cosh \zeta E(u_0,u_1)+\sinh \zeta P_1(u_0,u_1)\\ P_1\left( \overrightarrow{{\mathcal {L}}_{\zeta }u}(0) \right)&=\cosh \zeta P_1(u_0,u_1)+\sinh \zeta E(u_0,u_1)\\ P_j\left( \overrightarrow{{\mathcal {L}}_{\zeta }u}(0) \right)&=P_j\left( (u_0,u_1)\right) ,\quad j=\llbracket 2,N \rrbracket . \end{aligned}$$

This is exactly (A.2) and (A.3). To make these formal computation rigorous, we smoothen the nonlinearity and the initial data. Let $\chi \in C_0^{\infty }({\mathbb {R}}^N)$ such that $\chi (v)=1$ if $|v|\le 1$ and $\chi (v)=0$ if $|v|\ge 2$. For $\varepsilon >0$, let

$$\begin{aligned} F_{\varepsilon }(v)=\left( 1-\chi \left( \frac{v}{\varepsilon } \right) \right) )\chi \left( \varepsilon v \right) |v|^{\frac{4}{N-2}}v, \end{aligned}$$

and note that $F_{\varepsilon }\in C_0^{\infty }\left( {\mathbb {R}}^N \right)$. Let $\left( u_{0,\varepsilon },u_{1,\varepsilon } \right) \in \left( C_0^{\infty }({\mathbb {R}}^N) \right) ^2$ such that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left\| (u_{0,\varepsilon },u_{1,\varepsilon })-(u_0,u_1)\right\| _{{\mathcal {H}}}=0. \end{aligned}$$

Let $u_{\varepsilon }$ be the solution of

$$\begin{aligned} \left\{ \begin{aligned} \partial _t^2u_{\varepsilon }-\Delta u_{\varepsilon }=|u_{\varepsilon }|^{\frac{4}{N-2}}u_{\varepsilon },\\ \overrightarrow{u}_{\varepsilon \restriction t=0}=(u_{0,\varepsilon },u_{1,\varepsilon })\in {\mathcal {H}}. \end{aligned} \right. \end{aligned}$$

(A.17)

Note that $u_{\varepsilon }$ is global, $C^{\infty }$, and that for all t

$$\begin{aligned} {{\,\mathrm{supp}\,}}\vec {u}_{\varepsilon }(t)\subset \{|x|\le |t|+R_{\varepsilon }\}, \end{aligned}$$

where $R_{\varepsilon }$ is such that ${{\,\mathrm{supp}\,}}\vec {u}_{\varepsilon }(0)\subset \{|x|\le R_{\varepsilon }\}$. Let

$$\begin{aligned} f_{\varepsilon }(v)=\int _0^{v}F_{\varepsilon }(w)\,dw. \end{aligned}$$

The energy

$$\begin{aligned} E_{\varepsilon }(u_{\varepsilon })=\frac{1}{2}\int |\nabla u_{\varepsilon }|^2+\frac{1}{2}\int (\partial _t u_{\varepsilon })^2-\int f_{\varepsilon }(u_{\varepsilon }) \end{aligned}$$

and the momentum

$$\begin{aligned} P(u_{\varepsilon })=\int \nabla u_{\varepsilon }\partial _t u_{\varepsilon } \end{aligned}$$

are independent of time. The Lorentz transformation of $u_{\varepsilon }$, ${\mathcal {L}}_{\zeta }u_{\varepsilon }$ are solutions of (A.17) with $\left( C_0^{\infty }({\mathbb {R}}^N)\right) ^2$ initial data. Explicit computations (which are rigorous in this context) prove that

$$\begin{aligned} E_{\varepsilon }\left( \overrightarrow{{\mathcal {L}}_{\zeta }u_{\varepsilon }}(0) \right)&=\cosh \zeta E_{\varepsilon }(u_{0,{\varepsilon }},u_{1,{\varepsilon }})+\sinh \zeta P_1(u_{0,{\varepsilon }},u_{1,\varepsilon })\\ P_1\left( \overrightarrow{{\mathcal {L}}_{\zeta }u_{\varepsilon }}(0) \right)&=\cosh \zeta P_1(u_{0,\varepsilon },u_{1,\varepsilon })+\sinh \zeta E_{\varepsilon }(u_{0,\varepsilon },u_{1,\varepsilon })\\ P_j\left( \overrightarrow{{\mathcal {L}}_{\zeta }u_{\varepsilon }}(0) \right)&=P_j\left( (u_{0,\varepsilon },u_{1,\varepsilon } \right) ,\quad j=\llbracket 2,N \rrbracket . \end{aligned}$$

It remains to prove that if $|\varvec{\ell }|\le \eta _0$, then

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}E_{\varepsilon }\left( \overrightarrow{{\mathcal {L}}_{\zeta }u_{\varepsilon }}(0) \right) =E\left( \overrightarrow{{\mathcal {L}}_{\zeta }u}(0) \right) ,\quad \lim _{\varepsilon \rightarrow 0}P\left( \overrightarrow{{\mathcal {L}}_{\zeta }u_{\varepsilon }}(0) \right) =P\left( \overrightarrow{{\mathcal {L}}_{\zeta }u}(0) \right) . \end{aligned}$$

(A.18)

We first prove

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left\| \overrightarrow{{\mathcal {L}}_{\zeta }(u_{\varepsilon })}(0)-\overrightarrow{{\mathcal {L}}_{\zeta }(u)}(0)\right\| _{{\mathcal {H}}}=0. \end{aligned}$$

(A.19)

For this, we start by proving

$$\begin{aligned} \sup _{-\tau \le t\le \tau } \left\| \vec {u}(t)-\vec {u}_{\varepsilon }(t)\right\| _{{\mathcal {H}}} +\left\| (u-u_{\varepsilon })1\!\!1_{{\mathcal {E}}}\right\| _{L^{\frac{N+2}{N-2}}_tL^{\frac{2(N+2)}{N-2}}_x}\underset{\varepsilon \rightarrow 0}{\longrightarrow } 0. \end{aligned}$$

(A.20)

Denote by $F(u)=|u|^{\frac{4}{N-2}}u$, $\psi _{\varepsilon }(u)=\left( 1-\chi \left( \frac{u}{\varepsilon } \right) \right) \chi (\varepsilon u)$. Then

$$\begin{aligned} \partial _t^2(u-u_{\varepsilon })-\Delta (u-u_{\varepsilon })&=\psi _{\varepsilon }(u_{\varepsilon })\left( F(u)1\!\!1_{{\mathcal {E}}}-F(u_{\varepsilon }) \right) +\left( 1-\psi _{\varepsilon }(u_{\varepsilon })\right) F(u)1\!\!1_{{\mathcal {E}}} \\ \overrightarrow{u-u_{\varepsilon }}_{\restriction t=0}&= (u_0,u_1)-(u_{0,\varepsilon },u_{1,\varepsilon }). \end{aligned}$$

As a consequence, for all $t_0\ge 0$,

$$\begin{aligned}&\left\| (u-u_{\varepsilon })1\!\!1_{{\mathcal {E}}}\right\| _{L^{\frac{N+2}{N-2}}\left( [0,t_0),L^{\frac{2(N+2)}{N-2}}_x\right) }\\&\quad \lesssim \left\| (F(u)-F(u_{\varepsilon }))1\!\!1_{{\mathcal {E}}}\right\| _{L^1\left( [0,t_0),L^{2}_x\right) }+\left\| (1-\psi _{\varepsilon }(u_{\varepsilon }))F(u)1\!\!1_{{\mathcal {E}}}\right\| _{L^1\left( [0,t_0),L^2_x\right) }\\&\qquad +\left\| (u_0,u_1)-(u_{0,\varepsilon },u_{1,\varepsilon })\right\| _{{\mathcal {H}}}. \end{aligned}$$

We write

$$\begin{aligned} \left( 1-\psi _{\varepsilon }(u_{\varepsilon }) \right) F(u)1\!\!1_{{\mathcal {E}}} =\left( 1\!\!1_{|u-u_{\varepsilon }|<\frac{1}{2} |u|}+1\!\!1_{|u-u_{\varepsilon }|\ge \frac{1}{2} |u|}\right) \left( 1-\psi _{\varepsilon }(u_{\varepsilon }) \right) F(u)1\!\!1_{{\mathcal {E}}}. \end{aligned}$$

We have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} 1\!\!1_{|u-u_{\varepsilon }|<\frac{1}{2} |u|}\left( 1-\psi _{\varepsilon }(u_{\varepsilon }) \right) F(u)1\!\!1_{{\mathcal {E}}}=0\quad \text {a.e.} \end{aligned}$$

(A.21)

Indeed, if x is fixed, then

$$\begin{aligned} 1\!\!1_{|u-u_{\varepsilon }|<\frac{1}{2} |u|} \left( 1-\psi _{\varepsilon }(u_{\varepsilon }) \right) \le {\left\{ \begin{array}{ll} 0 &{} \text { if }\varepsilon \le |u_{\varepsilon }(x)|\le \frac{1}{\varepsilon }\\ \chi \left( \frac{|u(x)|}{2\varepsilon } \right) &{}\text { if } |u_{\varepsilon }(x)|\le \varepsilon \\ \left( 1-\chi \left( \frac{3}{2}\varepsilon |u(x)| \right) \right) &{}\text { if }|u_{\varepsilon }(x)|\ge \frac{1}{\varepsilon }, \end{array}\right. } \end{aligned}$$

where we have used that $\chi$ is decreasing. This obviously implies (A.21). As a consequence of (A.21), by the dominated convergence theorem

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\left\| 1\!\!1_{|u-u_{\varepsilon }|<\frac{1}{2} |u|} \left( 1-\psi _{\varepsilon }(u_{\varepsilon }) \right) F(u)1\!\!1_{{\mathcal {E}}}\right\| _{L^{\frac{N+2}{N-2}}_tL^{\frac{2(N+2)}{N-2}}}=0. \end{aligned}$$

On the other hand,

$$\begin{aligned} 1\!\!1_{|u-u_{\varepsilon }|\ge \frac{1}{2} |u|}\left( 1-\psi _{\varepsilon }(u_{\varepsilon }) \right) F(u)\lesssim |u|^{\frac{4}{N}}|u-u_{\varepsilon }|. \end{aligned}$$

Using Strichartz estimates and the equation satisfied by $u-u_{\varepsilon }$ we deduce that for all $t_0>0$,

$$\begin{aligned}&\left\| (u-u_{\varepsilon })1\!\!1_{{\mathcal {E}}}\right\| _{L^{\frac{N+2}{N-2}}\left( [0,t_0),L^{\frac{2(N+2)}{N-2}}_x\right) }\\&\quad \lesssim \left\| 1\!\!1_{{\mathcal {E}}}|u-u_{\varepsilon }| |u|^{\frac{4}{N}}\right\| _{L^1\left( [0,t_0),L^2_x \right) }+\left\| |u-u_{\varepsilon }|^{1+\frac{4}{N}}1\!\!1_{{\mathcal {E}}}\right\| _{L^1\left( [0,t_0),L^2_x \right) }+o(1),\quad \varepsilon \rightarrow 0. \end{aligned}$$

Since $u1\!\!1_{{\mathcal {E}}}\in L^{\frac{N+2}{N-2}}({\mathbb {R}},L^{\frac{2(N+2)}{N-2}})$, we obtain, combining with the same argument for negative times,

$$\begin{aligned} \lim _{\varepsilon \rightarrow \infty }\left\| (u-u_{\varepsilon })1\!\!1_{{\mathcal {E}}}\right\| _{L^{\frac{N+2}{N-2}} \left( {\mathbb {R}},L^{\frac{2(N+2)}{N-2}}\right) }=0. \end{aligned}$$

Going back to the equation satisfied by $u-u_{\varepsilon }$ and using Strichartz estimates, we obtain (A.20). By Claim A.4, we deduce (A.19).

In view of (A.19), the following property will imply (A.18):

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int f_{\varepsilon }\left( {\mathcal {L}}_{\zeta }(u_{\varepsilon })(0,x) \right) \,dx=\frac{N-2}{2N}\int \left| {\mathcal {L}}_{\zeta }(u)(0,x) \right| ^{\frac{2N}{N-2}}\,dx. \end{aligned}$$

Denote $w(x)= {\mathcal {L}}_{\zeta }(u)(0,x)$, $w_{\varepsilon }(x)={\mathcal {L}}_{\zeta }(u_{\varepsilon })(0,x)$, and $f(u)=\frac{N-2}{2N}|u|^{\frac{2N}{N-2}}$. Write

$$\begin{aligned}&\int f_{\varepsilon }\left( w_{\varepsilon }(x)\right) \,dx-\int f\left( w(x)\right) \,dx\\&\quad =\int f_{\varepsilon }\left( w_{\varepsilon }(x)\right) \,dx-\int f_{\varepsilon }\left( w(x)\right) \,dx+\int f_{\varepsilon }\left( w(x)\right) \,dx-\int f\left( w(x)\right) \,dx. \end{aligned}$$

We have $0\le f_{\varepsilon }\left( w\right) \le f\left( w\right)$ and $\lim _{\varepsilon \rightarrow 0} f_{\varepsilon }\left( w(x)\right) =f\left( w(x)\right) ,$ a.e., which implies

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int f_{\varepsilon }\left( w(x)\right) \,dx=\int f\left( w(x)\right) \,dx, \end{aligned}$$

by the dominated convergence theorem.

On the other hand,

$$\begin{aligned}&\left| f_{\varepsilon }\left( w_{\varepsilon }(x)\right) -f_{\varepsilon }\left( w(x)\right) \right| =\left| \int _{w(x)}^{w_{\varepsilon }(x)}F_{\varepsilon }(\sigma )\,d\sigma \right| \\&\quad \le \left| \int _{w(x)}^{w_{\varepsilon }(x)}F(\sigma )\,d\sigma \right| \le \left| F\left( w_{\varepsilon }(x) \right) +F\left( w(x) \right) \right| \,|w_{\varepsilon }(x)-w(x)|, \end{aligned}$$

where we have used that F is monotonic. By Hölder inequality,

$$\begin{aligned}&\int \left| f_{\varepsilon }(w_{\varepsilon }(x))-f_{\varepsilon }(w(x))\right| \,dx\\&\quad \lesssim \left( \int |w_{\varepsilon }(x)-w(x)|^{\frac{2N}{N-2}}\,dx \right) ^{\frac{N-2}{2N}} \left( \Vert w_{\varepsilon }\Vert ^{\frac{N+2}{N-2}}_{L^{\frac{2N}{N-2}}} +\Vert w\Vert ^{\frac{N+2}{N-2}}_{L^{\frac{2N}{N-2}}} \right) \underset{\varepsilon \rightarrow 0}{\longrightarrow }0. \end{aligned}$$

This concludes the proof.