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

TL;DR

This paper proves global existence and scattering for radial solutions of a defocusing energy sub-critical semi-linear wave equation in 3D with specific weighted initial data, using a transformation and integral estimates.

Contribution

It introduces a transformation that converts the original wave equation into a form with finite energy, enabling new global space-time estimates for scattering in the sub-critical case.

Findings

Solutions exist globally for all time.

Solutions scatter as time approaches infinity.

The transformation technique is effective for weighted initial data.

Abstract

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $p \in [3,5)$. We prove that if initial data $(u_0, u_1)$ are radial so that $\|\nabla u_0\|_{L^2 ({\mathbb R}^3; d\mu)}, \|u_1\|_{L^2 ({\mathbb R}^3; d\mu)} \leq \infty$, where $d \mu = (|x|+1)^{1+2\varepsilon}$ with $\varepsilon > 0$, then the corresponding solution $u$ must exist for all time $t \in {\mathbb R}$ and scatter. The key ingredients of the proof include a transformation $\mathbf{T}$ so that $v = \mathbf{T} u$ solves the equation $v_{\tau \tau} - \Delta_y v = - \left(\frac{|y|}{\sinh |y|}\right)^{p-1} e^{-(p-3)\tau} |v|^{p-1}v$ with a finite energy, and a couple of global space-time integral estimates regarding a solution $v$ as above.