Energy distribution of solutions to defocusing semi-linear wave equation in higher dimensional space

TL;DR

This paper investigates decay and scattering properties of solutions to the defocusing semi-linear wave equation in higher dimensions with finite energy initial data, establishing decay estimates and scattering results under certain conditions.

Contribution

It provides new decay estimates and scattering results for solutions with radially symmetric initial data in higher dimensions, extending previous understanding of the defocusing wave equation.

Findings

Decay estimates for solutions with specific initial decay rates

Scattering results for solutions with finite energy and weighted energy conditions

Extension of decay and scattering theory to higher-dimensional spaces

Abstract

The topic of this paper is a semi-linear, defocusing wave equation $u_{t t}-\Delta u=-|u|^{p-1} u$ in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite energy. We prove some decay estimates of the the solutions if initial data decay at a certain rate as the spatial variable tends to infinity. A combination of this property with a method of characteristic lines give a scattering result if the initial data satisfy $$E_{\kappa}\left(u_{0}, u_{1}\right)=\int_{\mathbb{R}^{d}}\left(|x|^{\kappa}+1\right)\left(\frac{1}{2}\left|\nabla u_{0}(x)\right|^{2}+\frac{1}{2}\left|u_{1}(x)\right|^{2}+\frac{1}{p+1}\left|u_{0}(x)\right|^{p+1}\right) d x<+\infty.$$ Here $\kappa=\frac{(2-d)p+(d+2)}{p+1}$.