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

TL;DR

This paper proves the global existence of smooth solutions for the critical semilinear wave equation with variable coefficients outside obstacles in three dimensions, extending results known for constant coefficients.

Contribution

It introduces a geometric multiplier approach and uses Riemannian geometry and Strichartz inequalities to establish global solutions in the variable coefficient setting.

Findings

Energy cannot concentrate at any point in space-time

Global existence of smooth solutions is proved

Method extends constant coefficient results to variable coefficients

Abstract

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 $(t,x)\in(0,\infty)\times\Omega$. 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.