Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven by Locally Lipschitz Lévy Noises

• Published in: Electronic Journal of Applied Mathematics Vol. 2; no. 1; pp. 40 - 63
• Main Authors: Li, Guofu, Yang, Xiulan, Zhou, Liguang, Wang, Yan
• Format: Journal Article
• Language: English
• Published: 25.03.2024
• ISSN: 2980-2474; 2980-2474
• DOI: 10.61383/ejam.20242165

This paper is concerned with weak pullback mean random attractor of reversible Selkov lattice systems defined on the entire integer set \(\mathbb{Z}\) driven by locally Lipschitz Lévy noises. Firstly, we formulate the stochastic lattice equations to an abstract system defined in the non-concrete space \(\ell^2\times\ell^2\) of square-summable sequences. Secondly, we establish the global well-posedness of the systems with locally Lipschitz diffusion terms. Under certain conditions, we show that the long-time dynamics can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space \(L^2(\Omega,\ell^2\times\ell^2)\). To overcome the difficulty caused by the drift and diffusion terms, we adopt a stopping time technique to prove the convergence of solutions in probability. The mean random dynamical systems theory proposed by Wang (J. Differ. Equ., 31:2177-2204, 2019) is used to deal with the difficulty caused by the nonlinear noise.