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Exponential mixing for the white-forced damped nonlinear wave equation. (English) Zbl 1304.35430
Summary: 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.

MSC:
 35L71 Second-order semilinear hyperbolic equations 35R60 PDEs with randomness, stochastic partial differential equations 37A25 Ergodicity, mixing, rates of mixing 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35L20 Initial-boundary value problems for second-order hyperbolic equations
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