Bounded Solutions to an Energy Subcritical Non-linear Wave Equation on R^3

Abstract

In this work we consider an energy subcritical semi-linear wave equation ($3<p<5$) \[ \partial_t^2 u - Δu = ϕ(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t){|}_{t=0}=(u_0,u_1)\in {\dot{H}}^{s_p}\times {\dot{H}}^{s_p-1}({\mathbb{R}}^3)$, where $s_p=3/2-2/(p-1)$ and the function $\varphi :{\mathbb{R}}^3\to [-1,1]$ is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation $-\Delta W=\varphi (x)|W{|}^{p-1}W$ has a nonzero radial solution $W\in C^2({\mathbb{R}}^3)\cap {\dot{H}}^{s_p}({\mathbb{R}}^3)$, any radial solution $u$ with a finite uniform upper bound on the critical Sobolev norm $\Vert (u(\cdot ,t),{\partial }_{t}u(\cdot ,t)){\Vert }_{{\dot{H}}^{s_p}\times {\dot{H}}^{s_p}({\mathbb{R}}^3)}$ for all $t$ in the maximal lifespan must be a global solution in time and scatter.

arXiv comment35 pages