Mean attractors of stochastic delay lattice [formula omitted]-Laplacian equations driven by superlinear noise in high-order product Bochner spaces

Abstract

This paper is concerned with the existence and uniqueness of mean random attractors of a class non-autonomous stochastic delay p-Laplacian lattice systems defined on a high-dimensional integer set Zd driven by a family infinite-dimensional superlinear noise. We first establish the global-in-time existence and uniqueness of the solutions in C([τ,∞),L2k(Ω,ℓ2(Zd)))∩Lq(Ω,Llocq((τ,∞),ℓq(Zd))) for any k⩾1 when the draft term has an arbitrary polynomial growth rate q>2 and the coefficient of the noise admits a superlinear growth order q̃∈[2,q). We then show that the mean random dynamical system generated by the solution operators has a unique weakly compact and weakly attracting mean random attractor in the high-order product Bochner space L2k(Ω,F;ℓ2(Zd))×L2k(Ω,F;L2k((−ϱ,0),ℓ2(Zd))), where ϱ is the time delay parameter. The dissipative property of the draft term is employed to carefully controlling the superlinear growth diffusion term. When k=1, our results are new even in the product Hilbert space L2(Ω,F;ℓ2(Zd))×L2(Ω,F;L2((−ϱ,0),ℓ2(Zd))). This work can be regard as a further study of mean attractors of stochastic p-Laplacian lattice systems in the works of Wang and Wang (2020) and Chen et al. (2023).