arXiv 1808.086562018-08math.AP

能量亚临界Wave Equation径向解的能量分布及其在散射理论中的应用

Energy Distribution of Radial Solutions to Energy Subcritical Wave Equation with an Application on Scattering Theory

Ruipeng Shen

备注：: 27 pages, 6 figures, a few errors fixed

摘要：

本文的研究对象是三维空间（$3\leq p<5$）中的半线性、能量亚临界、散焦波动方程$\partial_t^2 u - Δu = - |u|^{p -1} u$。我们将能量分解为向内和向外的能量，然后应用能流公式得到如下的能量渐近分布：除非溶液发生散射，否则其能量可分为两部分：“散射能”，它集中在光锥$|x|=|t|$周围，并以光速向无穷远处移动;“阻滞能”，当$|t|$较大时，它至少落后$|t|^β$一段距离。这里，$β$是小于$β_0(p) = \frac{2(p-2)}{p+1}$的任意常数。这个属性与一个更详细的版本的经典Morawetz估计的组合给出了一个散射的结果下，一个较弱的假设下的初始数据$(u_0,u_1)$比以前已知的结果。更准确地说，我们假设\[ \int_{{\mathbb R}^3} (|x|^κ+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u|^{p+1}\right) dx < +\infty. \]这里$κ>κ_0(p) =1-β_0(p) = \frac{5-p}{p+1}$是一个常数。

英文摘要：

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: "scattering energy" which concentrates around the light cone $|x|=|t|$ and moves to infinity at the light speed and "retarded energy" which is at a distance of at least $|t|^β$ behind when $|t|$ is large. Here $β$ is an arbitrary constant smaller than $β_0(p) = \frac{2(p-2)}{p+1}$. A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data $(u_0,u_1)$ than previously known results. More precisely, we assume \[ \int_{{\mathbb R}^3} (|x|^κ+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u|^{p+1}\right) dx < +\infty. \] Here $κ>κ_0(p) =1-β_0(p) = \frac{5-p}{p+1}$ is a constant.

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。

摘要：

英文摘要：

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