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arXiv 1508.04892math.AP
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R^3上能量次临界非线性Wave Equation的有界解

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

Ruipeng Shen

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35 pages
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摘要：

在这项工作中，我们考虑一个能量亚临界半线性波动方程（$3 < p < 5$）\[ \partial_t^2 u - Δu = ϕ(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \]与初始数据$(u,u_t)|_{t=0} = (u_0,u_1)\in \dot{H}^{s_p} \times \dot{H}^{s_p-1}({\mathbb R}^3)$，其中$s_p = 3/2 - 2/(p-1)$和函数$ϕ: {\mathbb R}^3 \rightarrow [-1,1]$是一个径向连续的功能与极限在无穷远。我们证明了，除非椭圆方程$-ΔW = ϕ(x) |W|^{p-1} W$有一个非零的径向解$W \in C^2 ({\mathbb R}^3) \cap \dot{H}^{s_p} ({\mathbb R}^3)$，任何径向解$u$的临界Sobolev范数$\|(u(\cdot,t), \partial_t u(\cdot,t))\|_{\dot{H}^{s_p}\times \dot{H}^{s_p}({\mathbb R}^3)}$的最大寿命的所有$t$上的有限的一致上界必须是在时间和分散的整体解决方案。

英文摘要：

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 $ϕ: {\mathbb R}^3 \rightarrow [-1,1]$ is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation $-ΔW = ϕ(x) |W|^{p-1} W$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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