Weak mean random attractors for nonautonomous stochastic parabolic equation with variable exponents

In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term.