A Semi-linear Energy Critical Wave Equation With Applications

Abstract

In this work we consider a semi-linear energy critical wave equation in ${\mathbb{R}}^d$ ($3\le d\le 5$) \[ \partial_t^2 u - Δu = \pm ϕ(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] 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 $\varphi \in C({\mathbb{R}}^d;(0,1])$ converges to zero as $|x|\to \infty$. We follow the same compactness-rigidity argument as Kenig and Merle applied on the Cauchy problem of the equation \[ \partial_t^2 u - Δu = |u|^{4/(d-2)} u \] and obtain a similar result when $\varphi$ 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.