A semi-linear energy critical wave equation with an application

In this paper we consider an energy critical wave equation (3≤d≤5, ζ=±1)∂t2u−Δu=ζϕ(x)|u|4/(d−2)u,(x,t)∈Rd×R with initial data (u,∂tu)|t=0=(u0,u1)∈H˙1×L2(Rd). Here ϕ∈C(Rd;(0,1]) converges as |x|→∞ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ∂t2u−Δu=|u|4/(d−2)u. Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case ζ=−1. While in the focusing case ζ=1 we give a criterion for global behaviour of the solutions, either scattering or finite-time blow-up when the energy is smaller than a certain threshold. As an application we give a similar criterion on the global behaviour of radial solutions to the focusing, energy critical shifted wave equation ∂t2v−(ΔH3+1)v=|v|4v on the hyperbolic space H3.