Scattering of solutions to the defocusing energy sub-critical semi-linear wave equation in 3D

Abstract

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation ${\partial }_t^{2}u-\Delta u=-|u{|}^{p-1}u$ in the 3-dimensional space with $p\in [3,5)$. We prove that if initial data $(u_0,u_1)$ are radial so that $\Vert \nabla u_0{\Vert }_{L^2({\mathbb{R}}^3;d\mu )},\Vert u_1{\Vert }_{L^2({\mathbb{R}}^3;d\mu )}\le \infty$, where $d\mu =(|x|+1{)}^{1+2\epsilon }$ with $\epsilon >0$, then the corresponding solution $u$ must exist for all time $t\in \mathbb{R}$ and scatter. The key ingredients of the proof include a transformation $\mathbf{T}$ so that $v=\mathbf{T}u$ solves the equation $v_{\tau \tau }-{\Delta }_{y}v=-{\left(\frac{|y|}{\sinh |y|}\right)}^{p-1}{e}^{-(p-3)\tau }|v{|}^{p-1}v$ with a finite energy, and a couple of global space-time integral estimates regarding a solution $v$ as above.

20 pages, 2 figures

References in corpus (1)

• Scattering for the radial 3d cubic wave equation