Limiting Behavior of Non-Autonomous Stochastic Reversible Selkov Lattice Systems Driven by Locally Lipschitz L\'{e}vy Noises

TL;DR

This paper studies the long-term behavior of non-autonomous stochastic Selkov lattice systems driven by Lévy noises, establishing the existence of unique pullback measure attractors and analyzing their properties under periodic forcing.

Contribution

It introduces a framework for analyzing the distributional dynamics of stochastic lattice systems with polynomial growth nonlinearities and proves the existence and periodicity of pullback measure attractors.

Findings

Existence of a unique pullback measure attractor for the system.

Pullback measure attractors are periodic under periodic external forcing.

Demonstrated upper semicontinuity of attractors as parameters tend to zero.

Abstract

This work investigates the long-term distributional behavior of the reversible Selkov lattice systems defined on the set $\mathbb{Z}$ and driven by locally Lipschitz \emph{L\'{e}vy noises}, which possess two pairs of oppositely signed nonlinear terms and whose nonlinear couplings can grow polynomially with any order $p \geq 1$. Firstly, based on the global-in-time well-posedness in $L^{2}(\Omega, \ell^2 \times \ell^2)$, we define a \emph{continuous} non-autonomous dynamical system (NDS) on the metric space $(\mathcal{P}_{2}(\ell^2 \times \ell^2), d_{\mathcal{P}(\ell^2 \times \ell^2)})$, where $d_{\mathcal{P}(\ell^2 \times \ell^2)}$ is the dual-Lipschitz distance on $\mathcal{P}(\ell^2 \times \ell^2)$, the space of probability measures on $\ell^2 \times \ell^2$. Specifically, we establish that this non-autonomous dynamical system admits a unique pullback measure attractor, characterized via measure-valued complete solutions and orbits in the sense of Wang (DOI.org/10.1016/j.jde.2012.05.015). Moreover, when the deterministic external forcing terms are periodic in time, we demonstrate that the pullback measure attractors are also periodic. We also study the upper semicontinuity of pullback measure attractors as $(\epsilon_1, \epsilon_2, \gamma_1, \gamma_2) \rightarrow (0, 0, 0, 0)$. The main difficulty in proving the pullback asymptotic compactness of the NDS in $(\mathcal{P}_{2}(\ell^2 \times \ell^2), d_{\mathcal{P}(\ell^2 \times \ell^2)})$ is caused by the lack of compactness in infinite-dimensional lattice systems, which is overcome by using uniform tail-ends estimates. And the inherent structure of the Selkov system precludes the possibility of any unidirectional dissipative influence arising from the interaction between the two coupled equations, thereby obstructing the emergence of a dominant energy-dissipation mechanism along a single directional pathway.