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.

中文翻译：

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R3 上能量次临界非线性波动方程的有界解

在这项工作中，我们考虑一个能量次临界半线性波动方程 ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \ qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] 初始数据 $(u,u_t)|_{t=0} = (u_0,u_1)\in \dot {H}^{s_p} \times \dot{H}^{s_p-1}({\mathbb R}^3)$，其中 $s_p = 3/2 - 2/(p-1)$ 和函数$\phi: {\mathbb R}^3 \rightarrow [-1,1]$ 是一个无限远的径向连续函数。我们证明除非椭圆方程 $-\Delta W = \phi(x) |W|^{p-1} W$ 有一个非零径向解 $W \in C^2 ({\mathbb R}^3) \cap \dot{H}^{s_p} ({\mathbb R}^3)$，在临界 Sobolev 范数 $\|(u(\cdot,t) 上具有有限统一上限的任何径向解 $u$ , \partial_t u(\cdot,