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.
Categories: math.AP