Uniform stabilization of the wave equation on compact manifolds and locally distributed damping – a sharp result

Let (M,g)(M,g) be an n  -dimensional (n⩾2n⩾2) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C∞C∞. This paper is concerned with the study of the wave equation on (M,g)(M,g) with locally distributed damping, described byutt−Δgu+a(x)g(ut)=0on M×]0,∞[,u=0 on∂M×]0,∞[, where ∂M represents the boundary of M and the last condition is dropped when M   is boundaryless. Let ϵ>0ϵ>0. We prove that there exist an open subset V⊂MV⊂M and a smooth function f:M→Rf:M→R such that meas(V)⩾meas(M)−ϵmeas(V)⩾meas(M)−ϵ, Hessf≈gHessf≈g on V   and infx∈V|∇f(x)|>0infx∈V|∇f(x)|>0. This function f   is used in order to prove that if a(x)⩾a0>0a(x)⩾a0>0 on an open subset M*⊂MM*⊂M that contains M\VM\V and if g   is a monotonic increasing function such that k|s|⩽|g(s)|⩽K|s|k|s|⩽|g(s)|⩽K|s| for all |s|⩾1|s|⩾1, then uniform and optimal decay rates of the energy hold. Therefore, given an arbitrary ϵ>0ϵ>0, uniform and optimal decay rates of the energy hold if the damping is effective in a well-chosen open subset with volume less than ϵ.