# American Institute of Mathematical Sciences

December  2014, 3(4): 645-670. doi: 10.3934/eect.2014.3.645

## Exponential mixing for the white-forced damped nonlinear wave equation

 1 Department of Mathematics, CNRS UMR 8088, University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95300 Cergy-Pontoise, France

Received  April 2014 Revised  September 2014 Published  October 2014

The paper is devoted to studying the stochastic nonlinear wave (NLW) equation $$\partial_t^2 u + \gamma \partial_t u - \triangle u + f(u)=h(x)+\eta(t,x)$$ in a bounded domain $D\subset\mathbb{R}^3$. The equation is supplemented with the Dirichlet boundary condition. Here $f$ is a nonlinear term, $h(x)$ is a function in $H^1_0(D)$ and $\eta(t,x)$ is a non-degenerate white noise. We show that the Markov process associated with the flow $\xi_u(t)=[u(t),\dot u (t)]$ has a unique stationary measure $\mu$, and the law of any solution converges to $\mu$ with exponential rate in the dual-Lipschitz norm.
Citation: Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645
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