Decay Estimates for Nonradiative Solutions of the Energy-Critical Focusing Wave Equation

Abstract

We consider the energy-critical focusing wave equation in space dimension $N\ge 3$. The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation, behaves asymptotically as a sum of modulated Ws, decoupled by the scaling, and a radiation term. A nonradiative solution of the equation is by definition a solution of which energy in the exterior $\{|x|>|t|\}$ of the wave cone vanishes asymptotically as $t\rightarrow +\infty$ and $t\rightarrow -\infty$. In our previous work [9], we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, 0 and $\pm W$. This was crucial in the proof of soliton resolution in [9]. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behavior as $r\rightarrow \infty$. We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag, and the second author in [15]. We also study the propagation of the support of nonzero radial solutions with compactly supported initial data and prove that these solutions cannot be nonradiative.

Appendix A: Sequences with Geometric Growth

In this appendix, we prove Claim 3.9. In all the proof, C (respectively, $\varepsilon$) will denote a large (respectively small) constant that may change from line to line and is allowed to depend on r, q, $c_0$, and $\beta$ but not on the other parameters.

We first assume $c_0=0$. Thus, we have

$$\begin{aligned} \forall n\ge 0,\quad \mu _{n+1}\le q\mu _n+\nu _0r^n. \end{aligned}$$

By a straightforward induction, we obtain

$$\begin{aligned} \forall n\ge 0,\quad \mu _{n}\le q^n\mu _0+\nu _0 r^{n-1}\sum _{j=0}^{n-1} \left( \frac{q}{r} \right) ^{j}. \end{aligned}$$

(A.1)

If $q\ne r$ we deduce

$$\begin{aligned} \mu _n\le q^n \mu _0+\nu _0r^{n-1}\frac{1-\left( \frac{q}{r} \right) ^n}{1-\frac{q}{r}}. \end{aligned}$$

In the case where $q<r$, this yields

$$\begin{aligned} \mu _n\le q^n\mu _0+\frac{\nu _0}{1-\frac{q}{r}} r^{n-1}\le C(\mu _0+\nu _0)r^n. \end{aligned}$$

When $q>r$, we have

$$\begin{aligned} \mu _n\le q^n\mu _0+\nu _0r^{n-1}\frac{\left( \frac{q}{r} \right) ^n}{\frac{q}{r}-1}\le C(\mu _0+\nu _0)q^n. \end{aligned}$$

Finally, in the case $q=r$, the inequality (A.1) is

$$\begin{aligned} \mu _n\le r^n\mu _0+\nu _0 n r^n\le C(\mu _0q^n+n\nu _0r^n). \end{aligned}$$

We next treat the general case. We first note that the assumptions (3.13) and (3.14) imply

$$\begin{aligned} \mu _{n+1} \le \left( q+c_0\varepsilon ^{\beta -1}\right) \mu _n+\nu _0r^n. \end{aligned}$$

(A.2)

If $q<r$, we choose $\varepsilon$ so small, so that $(q+c_0\varepsilon ^{\beta -1})\le \frac{q+r}{2}<r$. Using the case $c_0=0$ treated previously, we obtain

$$\begin{aligned} \mu _k\le C(\mu _0+\nu _0)r^k. \end{aligned}$$

If $q\ge r$, we have $q+c_0\varepsilon ^{\beta -1}>r$, and (A.2) implies, using the case $c_0=0$,

$$\begin{aligned} \mu _n\le C \left( q+c_0\varepsilon ^{\beta -1}\right) ^{n}(\mu _0+\nu _0). \end{aligned}$$

(A.3)

Plugging this into (3.13), we deduce

$$\begin{aligned} \mu _{n+1}\le q\mu _n+\nu _0r^n+ C^{\beta }(\mu _0+\nu _0)^{\beta }c_0\left( q+c_0\varepsilon ^{\beta -1} \right) ^{n\beta }. \end{aligned}$$

We choose $\varepsilon$ small, so that $q'=(q+c_0\varepsilon ^{\beta -1})^{\beta }<q$, which is possible since $\beta >1$ and $q<1$. We deduce

$$\begin{aligned} \mu _{n+1}\le q\mu _n+C (\nu _0+\mu _0) (\max (q',r))^n. \end{aligned}$$

We use again the case $c_0=0$. If $r<q$, we obtain (3.15). If $q=r$, we have $\max (q',r)=r=q$ and (3.16) follows. $\square$