Dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise

This paper is concerned with the well-posedness as well as long-term dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise. We will apply a specific method to solve stochastic Ginzburg–Landau equations, known as the variational approach. We prove the existence and uniqueness of the solutions by assuming that the coefficients satisfy certain monotonicity assumptions. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. At the same time, the existence of invariant measures for the stochastic Ginzburg–Landau equations is also established.