EXPONENTIAL MIXING FOR THE WHITE - FORCED DAMPED NONLINEAR WAVE EQUATION

The paper is devoted to studying the stochastic nonlinear wave (NLW) equation partial derivative(2)(t) u + gamma partial derivative(t) u - Delta(u) + f(u) = h(x) + eta (t, x) in a bounded domain D subset of R-3. The equation is supplemented with the Dirichlet boundary condition. Here f is a nonlinear term, h(x) is a function in H-0(1)(D) and eta(t,x) is a non-degenerate white noise. We show that the Markov process associated with the flow xi(u) (t) = [u(t), <(u) (t)] has a unique stationary measure mu, and the law of any solution converges to mu with exponential rate in the dual-Lipschitz norm.