Random attractors for stochastic time-dependent damped wave equation with critical exponents

Qingquan Chang; Dandan Li; Chunyou Sun

doi:10.3934/dcdsb.2020033

Article Contents

2020, Volume 25, Issue 7: 2793-2824. Doi: 10.3934/dcdsb.2020033

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Random attractors for stochastic time-dependent damped wave equation with critical exponents

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

^* Corresponding author: Chunyou Sun
^* Corresponding author: Chunyou Sun

Received: April 30, 2019

Revised: August 31, 2019

Early access: April 2020

Published: July 2020

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Abstract

We study the asymptotic behavior of solutions of a stochastic time-dependent damped wave equation. With the critical growth restrictions on the nonlinearity $ f $ and the time-dependent damped term, we prove the global existence of solutions and characterize their long-time behavior. We show the existence of random attractors with finite fractal dimension in $ H^1_0(U)\times L^2(U) $. In particular, the periodicity of random attractors is also obtained with periodic force term and coefficient function. Furthermore, we construct the pullback random exponential attractors.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 35B40, 60H15.

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