Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay

Abstract

The well-posedness and asymptotic dynamics of second-order-in-time stochastic evolution equations with state-dependent delay is investigated. This class covers several important stochastic PDE models arising in the theory of nonlinear plates with additive noise. We first prove well-posedness in a certain space of functions which are \begin{document}$C^1$\end{document} in time. The solutions constructed generate a random dynamical system in a \begin{document}$C^1$\end{document} -type space over the delay time interval. Our main result shows that this random dynamical system possesses compact global and exponential attractors of finite fractal dimension. To obtain this result we adapt the recently developed method of quasi-stability estimates to the random setting.