Asymptotic Stability of the Wave Equation on Compact Surfaces and Locally Distributed Damping - a Sharp Result

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by utt −∆Mu+ a(x) g(ut) = 0 on M× ]0,∞[ , where M ⊂ R is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced on M by R, we prove that for each ǫ > 0, there exist an open subset V ⊂ M and a smooth function f : M → R such that meas(V ) ≥ meas(M)− ǫ, Hessf ≈ g on V and inf x∈V |∇f(x)| > 0. In addition, we prove that if a(x) ≥ a0 > 0 on an open subset M∗ ⊂ M which contains M\V and if g is a monotonic increasing function such that k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then uniform and optimal decay rates of the energy hold.