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Abstract. This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by 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.

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy-Ventcel boundary conditions:where is a bounded domain of R n (n ≥ 2) having a smooth boundary := ∂ , such that = 0 ∪ 1 with 0 , 1 being closed and disjoint. It is known that if a(x) = 0 then the uniform exponential stability never holds even if a linear frictional feedback is applied to the entire boundary of the domain [see, for instance, Hemmina (ESAIM, Control Optim Calc Var 5:591-622, 2000, Thm. 3.1)]. Let f : → R be a smooth function; define ω 1 to be a neighbourhood of 1 , and subdivide the boundary 0 into two parts: * 0 = {x ∈ 0 ; ∂ ν f > 0} and 0 \ * 0 . Now, let ω 0 be a neighbourhood of * 0 . We prove that if a(x) ≥ a 0 > 0 on the open subset ω = ω 0 ∪ ω 1 and if g is a monotone increasing function satisfying k|s| ≤ |g(s)| ≤ K |s| for all |s| ≥ 1, then the energy of the system decays uniformly at the rate quantified by the solution to a certain nonlinear ODE dependent on the damping [as in Lasiecka and Tataru (Differ Integral Equ 6:507-533, 1993)].

Abstract. This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by3 is a smooth (of class C 3 ) oriented embedded compact surface without boundary, such that M = M0 ∪ M1, whereHere, m(x) := x − x 0 , (x 0 ∈ R 3 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).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.

Abstract. Invariance entropy for the action of topological semigroups acting on metric spaces is introduced. It is shown that invariance entropy is invariant under conjugations and a lower bound and upper bounds of invariance entropy are obtained. The special case of control systems is discussed.