Global existence of solutions of the critical semilinear wave equations with variable coefficients outside obstacles

Abstract

In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coefficients case, we show that the energy cannot concentrate at any point (t, x) ∈ (0,∞) × $\bar \Omega$. For that purpose, following Ibrahim and Majdoub’s paper in 2003, we use a geometric multiplier similar to the well-known Morawetz multiplier used in the constant coefficients case. We then use the comparison theorem from Riemannian geometry to estimate the error terms. Finally, using the Strichartz inequality as in Smith and Sogge’s paper in 1995, we confirm the global existence.