Uniform Stabilization of the Wave Equation on Compact Surfaces and Locally Distributed Damping

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 (of class C) oriented embedded compact surface without boundary, such that M = M0 ∪M1, where M1 := {x ∈ M;m(x) · ν(x) > 0} and M0 = M\M1. Here, m(x) := x − x, (x ∈ R fixed) and ν is the exterior unit normal vector field of M. For i = 1, . . . , k, assume that there exist open subsets M0i ⊂ M0 of M with smooth boundary ∂M0i such that M0i are umbilical, or more generally, that the principal curvatures k1 and k2 satisfy |k1(x)− k2(x)| < εi (εi considered small enough) for all x ∈ M0i. Moreover suppose that the mean curvature H of each M0i is non-positive (i.e. H ≤ 0 on M0i for every i = 1, . . . , k). If a(x) ≥ a0 > 0 on an open subset M∗ ⊂ M that contains M\∪ki=1 M0i and if g is a monotonic increasing function such that k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then uniform decay rates of the energy hold.