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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 $|x| \rightarrow \infty$. Wefollow the same compactness-rigidity argument as Kenig and Merle applied on theCauchy problem of the equation \[  \partial_t^2 u - \Delta u = |u|^{4/(d-2)} u \] and obtain a similar resultwhen $\phi$ satisfies some technical conditions. In the defocusing case weprove 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 globalbehaviour of the solutions, either scattering or finite-time blow-up, accordingto their initial data when the energy is smaller than a certain threshold.

A Semi-linear Energy Critical Wave Equation With Applications