$\mathcal{D}$-Pullback Measure Attractors of McKean-Vlasov Stochastic Lattice Wave Equations

Abstract

This paper deals with the pullback measure attractor for a non-autonomous distribution-dependent stochastic lattice wave equation with nonlinear noise on $\mathbb{Z}$. We establish well-posedness in the energy space $\ell^{2}\times\ell^{2}$ and show that the solution generates a continuous non-autonomous dynamical system on $\mathcal{P}_{4}(\ell^{2}\times\ell^{2})$, which possesses a unique pullback measure attractor $\mathcal{A}$. Furthermore, we prove the convergence of the attractor $\mathcal{A}^{\varepsilon}$ as $\varepsilon\to 0^{+}$ to the attractor of the corresponding distribution-independent limit system.