Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

Renhai Wang; Bixiang Wang

doi:10.3934/dcdsb.2020019

Article Contents

2020, Volume 25, Issue 7: 2461-2493. Doi: 10.3934/dcdsb.2020019

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Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

Renhai Wang^1, , and
Bixiang Wang^2,

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

^* Corresponding author: rwang-math@outlook.com (Renhai Wang)
^* Corresponding author: rwang-math@outlook.com (Renhai Wang)

Received: May 31, 2019

Revised: July 31, 2019

Early access: April 2020

Published: July 2020

The first author is supported by the Innovation Project of Chongqing grant CYB18115 and the China Scholarship Council(CSC No.201806990064)

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Abstract

This paper is concerned with the global existence and random dynamics of non-autonomous stochastic second-order lattice systems driven by infinite-dimensional nonlinear noise defined on higher-dimensional integer sets. We first show the existence and uniqueness of mean square solutions to the equations when the nonlinear drift term has a polynomial growth of arbitrary order and the diffusion term is locally Lipschitz continuous. We then prove that the mean random dynamical system associated with the solution operator possesses a unique tempered weak pullback mean random attractor in a Bochner space under certain conditions. We finally establish the existence of invariant measures for the stochastic systems in $ \ell^2\times\ell^2 $ by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 37B55, 35B41, 35B40.

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