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

TL;DR

This paper investigates the behavior of solutions to an energy subcritical nonlinear wave equation in three dimensions, establishing conditions under which solutions are global and scatter, based on the nonexistence of certain elliptic solutions.

Contribution

It proves that solutions with bounded critical Sobolev norm are global and scatter unless a specific elliptic equation admits a nonzero radial solution.

Findings

Solutions are global and scatter if no elliptic solution exists.

Bounded Sobolev norm implies global existence under certain conditions.

Characterization of solutions based on elliptic equation solutions.

Abstract

In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] 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 $\phi: {\mathbb R}^3 \rightarrow [-1,1]$ is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation $-\Delta W = \phi(x) |W|^{p-1} W$ 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 $\|(u(\cdot,t), \partial_t u(\cdot,t))\|_{\dot{H}^{s_p}\times \dot{H}^{s_p}({\mathbb R}^3)}$ for all $t$ in the maximal lifespan must be a global solution in time and scatter.