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

Igor Chueshov; Peter E. Kloeden; Meihua Yang

doi:10.3934/dcdsb.2018139

Article Contents

2018, Volume 23, Issue 3: 991-1009. Doi: 10.3934/dcdsb.2018139

This issue Previous Article In memoriam Igor D. Chueshov September 23, 1951 - April 23, 2016 Next Article Robustness of time-dependent attractors in H¹-norm for nonlocal problems

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

1.
Department of Mechanics and Mathematics, Kharkov National University, 61077, Kharkov, Ukraine
2.
School of Mathematics & Statistics, Huazhong University of Science & Technology, Wuhan 430074, China

Received: August 31, 2016

Revised: December 31, 2016

Early access: February 2018

Published: June 2018

Partially supported by the Chinese NSF grant no. 1157112 and NCET-12-0204, and the Spanish Ministerio de Economía y Competitividad project project MTM2015-63723-P

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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 $C^1$ in time. The solutions constructed generate a random dynamical system in a $C^1$-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.

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Mathematics Subject Classification: Primary: 60H15, 35K90.

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