Global Well-Posedness, Mean Attractors and Invariant Measures of Generalized Reversible Gray-Scott Lattice Systems Driven by Nonlinear Noise

The main objective of our investigation is to study the global well-posedness as well as stochastic dynamics of a class of generalized reversible Gray-Scott lattice systems (RGSLSs) driven by nonlinear white noise. Compared with the classical stochastic RGSLSs considered in the literature, the generalized stochastic RGSLSs have two significant features: the coupled drift terms have polynomial growth of arbitrary (not cubic) orders, and the diffusion coefficients of the noise are locally Lipschitz. The global well-posedness and existence of mean random attractors are established for the nonautonomous RGSLSs. The existence and limit behavior of invariant probability measures for the autonomous RGSLSs are studied by the idea of uniform tail-estimates due to Wang (Physica D 128:41-52, 1999) and a scaling method used in Gu and Xiang (Appl Math Comput 225:387-400, 2013) in order to surmount the difficulties caused by the noncompactness in infinite-dimensional lattice systems and the coefficient barrier in the z-equation. These results are new even when the arbitrary growth rate of the draft term reduces to the cubic growth.