Energy distribution of solutions to defocusing semi-linear wave equation in two dimensional space

Abstract

We consider finite-energy solutions to the defocusing nonlinear wave equation in two dimensional space. We prove that almost all energy moves to the infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. These behaviours resemble those of free waves. We also prove some decay estimates of the solutions if the initial data decay at a certain rate as the spatial variable tends to infinity. As an application, we prove a couple of scattering results for solutions whose initial data are in a weighted energy space. Our assumption on decay rate of initial data is weaker than previous known scattering results.

Appendix

In this appendix we give a brief proof of a Morawetz estimate for solutions to 2D wave equation. This kind of Morawetz estimates were first introduced by Nakanishi. For convenience we use the same notation as in Nakanishi [31]:

$$\begin{aligned}&\lambda = \sqrt{t^2+r^2};&\Theta = \frac{(-t,x)}{\lambda };&g = \frac{d-1}{2\lambda } + \frac{t^2-r^2}{2\lambda ^3};&\\&m_h = \Theta \cdot (u_t, \nabla u) + ug;&l(u) = \frac{|\nabla u|^2}{2} - \frac{|u_t|^2}{2} + \frac{|u|^{p+1}}{p+1};&\Box = \partial _t^2 - \Delta ;&\end{aligned}$$

and $(\partial _0, \partial _1, \cdots , \partial _d) = (-\partial ^0, \partial ^1, \cdots , \partial ^d) = (\partial _t, \nabla )$. Then we have an identity

A basic calculation shows that

$$\begin{aligned} \Box g = \frac{(d-3)(d+3)}{2\lambda ^3} + 3(d-1)\frac{t^2-r^2}{\lambda ^5}+15\frac{(t^2-r^2)^2}{2\lambda ^7}. \end{aligned}$$

Thus we may integrate in the region ${{\mathbb {R}}}^2 \times [1,T]$ and obtain

(15)

In order to deal with the terms involving $|u|^2$, we need to apply Hölder’s inequality

$$\begin{aligned} \int _{{{\mathbb {R}}}^2} \frac{|u(x,t)|^2}{\lambda ^2} dx \le \left( \int _{{{\mathbb {R}}}^2} |u|^{p+1} dx\right) ^{\frac{2}{p+1}} \left( \int _{{{\mathbb {R}}}^2} \lambda ^{-\frac{2(p+1)}{p-1}} dx\right) ^{\frac{p-1}{p+1}} \lesssim _p t^{-\frac{4}{p+1}} E^{\frac{2}{p+1}}. \end{aligned}$$

Next we observe the facts $|g_t| \lesssim \lambda ^{-2}$, $|\Box g| \lesssim \lambda ^{-3}$ and obtain

$$\begin{aligned} \left| \int _{{{\mathbb {R}}}^2} \left( -m_h u_t+\frac{t}{\lambda } l(u) + \frac{|u|^2}{2} g_t\right) dx\right|&\lesssim _p \int _{{{\mathbb {R}}}^2} \left( |\nabla u|^2 + |u_t|^2 + |u|^{p+1} + \frac{|u|^2}{\lambda ^2} \right) dx\\&\lesssim _p E + E^{\frac{2}{p+1}}, \end{aligned}$$

and

$$\begin{aligned} \int _{1}^T \int _{{{\mathbb {R}}}^2} \frac{|u|^2}{2} |\Box g| dx dt \le \int _1^T \int _{{{\mathbb {R}}}^2} \frac{|u|^2}{t\lambda ^2} dx dt \lesssim _p E^{\frac{2}{p+1}} \int _1^T t^{-1-\frac{4}{p+1}} dt \lesssim _p E^{\frac{2}{p+1}}. \end{aligned}$$

We plug these upper bound in (15) and conclude