Scattering of solutions to the defocusing energy sub-critical semi-linear wave equation in 3D

2015/12/02 by Shen, Ruipeng
#35L05 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1512.00705

Abstract

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation ∂_t² u - Δu = - |u|^{p -1} u in the 3-dimensional space with p ∈ [3,5). We prove that if initial data (u₀, u₁) are radial so that ‖∇ u₀‖_L² (\mathbb R³; dμ), ‖u₁‖_L² (\mathbb R³; dμ) ≤ ∞, where d μ= (|x|+1)^1+2ε with ε > 0, then the corresponding solution u must exist for all time t ∈ \mathbb R and scatter. The key ingredients of the proof include a transformation T so that v = T u solves the equation v_ττ - Δ_y v = - ((|y|)/(\sinh |y|))^p-1 e^-(p-3)τ |v|^p-1v with a finite energy, and a couple of global space-time integral estimates regarding a solution v as above.

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