Exponential Decay of the Local Energy for Solutions of the Damped Critical Wave Equation Outside the Union of Two Strictly Convex Obstacles

This paper is concerned with the defocusing energy-critical wave equation with a localized semilinearity and damping, set in the exterior of two disjoint, strictly convex obstacles in R3. The trapping nature of this geometry, due to a periodic ray bouncing between the obstacles, is the main obstruction to decay. Under an exterior geometric control condition, the local energy is shown to decay exponentially, uniformly over data supported in a fixed ball with energy below any prescribed level. The rate is exactly that of the damped linear flow, and hence independent of the data and of the energy level; only the constant depends on the latter. The approach relies on Lafontaine’s lossless Strichartz estimates, a Lax–Phillips and microlocal defect measure analysis of the damped linear flow, and a Duhamel–Gronwall scheme.