Exponential mixing for the white-forced damped nonlinear wave equation

Davit Martirosyan

doi:10.3934/eect.2014.3.645

Article Contents

2014, Volume 3, Issue 4: 645-670. Doi: 10.3934/eect.2014.3.645

This issue Previous Article On a linear problem arising in dynamic boundaries Next Article On a class of elliptic operators with unbounded diffusion coefficients

Exponential mixing for the white-forced damped nonlinear wave equation

Davit Martirosyan^1,

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

Received: April 2014

Revised: September 2014

Published: December 2014

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Abstract

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.

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Mathematics Subject Classification: 35L70, 35R60, 37A25, 60H15.

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