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

Let ( M , g ) be an n-dimensional ( n ⩾ 2 ) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C ∞ . This paper is concerned with the study of the wave equation on ( M , g ) with locally distributed damping, described by u t t − Δ g u + a ( x ) g ( u t ) = 0 on 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 > 0 . We prove that there exist an open subset V ⊂ M and a smooth function f : M → R such that meas ( V ) ⩾ meas ( M ) − ϵ , Hess f ≈ g on V and inf x ∈ V | ∇ f ( x ) | > 0 . This function f is used in order to prove that if a ( x ) ⩾ a 0 > 0 on an open subset M * ⊂ M that 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. Therefore, given an arbitrary ϵ > 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.