Global Existence of the Critical Semilinear Wave Equations with Variable Coefficients Outside Obstacles

In this paper, we consider exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove global existence of smooth solutions. Similar to the constant coefficients case, we show that the energy cannot concentrate at any point $(t,x)\in (0,\mathrm{\infty })\times \mathrm{\Omega }$. For that purpose, following Ibrahim and Majdoub \cite{Ibrahim}, we use a geometric multiplier close to the well-known Morawetz multiplier used in the constant coefficients case. Then we use comparison theorem from Riemannian Geometry to estimate the error terms. Finally, using Strichartz inequality as in Smith and Sogge \cite{Sogge}, we get the global existence.