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

2021/06/26 by Li, Liang, Shen, Ruipeng
#35L05 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.13994

Abstract

The topic of this paper is a semi-linear, defocusing wave equation u_{t t}-Δu=-|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_κ(u₀, u₁)=∫_ℝ^d(|x|^κ+1)((1)/(2)|∇ u₀(x)|²+(1)/(2)|u₁(x)|²+(1)/(p+1)|u₀(x)|^p+1) d x

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