A Semi-linear Energy Critical Wave Equation With Applications

TL;DR

This paper studies a semi-linear energy critical wave equation with a spatially decaying coefficient, establishing scattering and blow-up results using a compactness-rigidity approach for both defocusing and focusing cases.

Contribution

It extends the Kenig-Merle method to a semi-linear wave equation with a spatially vanishing coefficient, providing new scattering and blow-up criteria.

Findings

Solutions scatter in the defocusing case for all initial data.

In the focusing case, solutions either scatter or blow up depending on initial energy.

The approach adapts the compactness-rigidity method to variable coefficient equations.

Abstract

In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - \Delta u = \pm \phi(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data $(u, \partial_t u)|_{t=0} = (u_0,u_1) \in \dot{H}^1 \times L^2 ({\mathbb R}^d)$. Here the function $\phi \in C({\mathbb R}^d; (0,1])$ converges to zero as $|x| \rightarrow \infty$. We follow the same compactness-rigidity argument as Kenig and Merle applied on the Cauchy problem of the equation \[ \partial_t^2 u - \Delta u = |u|^{4/(d-2)} u \] and obtain a similar result when $\phi$ satisfies some technical conditions. In the defocusing case we prove that the solution scatters for any initial data in the energy space $\dot{H}^1 \times L^2$. While in the focusing case we can determine the global behaviour of the solutions, either scattering or finite-time blow-up, according to their initial data when the energy is smaller than a certain threshold.