Local energy decay for linear wave equations with variable coefficients

Abstract

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561–568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata–Matsuyama [ L 2 -behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33–42] and Todorova–Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489].