Exponential Stabilization of the Wave Equation on Hyperbolic Spaces with Nonlinear Locally Distributed Damping

摘要

In this article, we consider the wave equation on hyperbolic spaces H-n(n >= 2) with nonlinear locally distributed damping as follow: {u(tt) - Delta(g)u + a(x)g(u(t)) = 0 (x, t) is an element of H-n x (0,+infinity),(1) u(x, 0) = u(0)(x), u(0)(x, 0) = u(1)(x) x is an element of H-n. It is well-known that the energy of the system (1) is of polynomial decay in the Euclidean space. However, on hyperbolic spaces, owing to the following inequality integral(Hn) u(2)dx(g) <= C integral(Hn) |del(g)u|(2)(g)dx(g), f or u is an element of H-1 (H-n), (2) we prove the exponential stabilization of the wave equation by multiplier methods and compactness-uniqueness arguments.