Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

Abstract

This paper is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is established by using the Faedo--Galerkin approximation and an idea analogous to Da Prato and Zabczyk [Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992]. Second, we show the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the steady-state solution of the deterministic problem is also a solution of the stochastic one or when this does not happen. Next, to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation of the noise given by Wong and Zakai's technique using the so called colored noise. For this model, we can use the power of the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the cases of additive and multiplicative noise, it is proved that the Wong--Zakai approximation models possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem.