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In this work we consider an energy subcritical semi-linear wave equation ($3< p < 5$) \[  \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbbR}^3 \times {\mathbb R} \] 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 radialcontinuous function with a limit at infinity. We prove that unless the ellipticequation $-\Delta W = \phi(x) |W|^{p-1} W$ has a nonzero radial solution $W \inC^2 ({\mathbb R}^3) \cap \dot{H}^{s_p} ({\mathbb R}^3)$, 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 aglobal solution in time and scatter.

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