Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise

Abstract

This paper is concerned with the asymptotic behavior of solutions for non-autonomous stochastic fractional complex Ginzburg-Landau equations driven by multiplicative noise with α ∈ (0, 1). We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions, which is slightly different from the deterministic fractional case with α ∈ (1 2 , 1) and the real fractional case with α ∈ (0, 1). Consequently, we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain. At last, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero.

Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

“…In this following, we will prove this theorem by the Galerkin method and compactness argument, see [9,28].…”

“…The global existence and long time behavior of Ginzburg-Landau equation were studied in [10,17,30]. The random attractors of stochastic Ginzburg-Landau equations were got in [29,28,31,44].…”

“…However, the existence of such a random dynamical system is unknown in general for a nonlinear function h in (1)(see, e.g., [11,15]), which results in a serious challenge for studying the pathwise dynamics of the equation. At present, the existence of random attractors for (1) has been established only when h is u or independent of u (see, e.g., [29,28,31,44]). In the nonlinear case, some progress has been made for a class of stochastic PDEs driven by a fractional Brownian motion by using rough path analysis, see Gao et al [14].…”

“…By a standard procedure (see [28,58]), we can check that χ1 = f (u)+G δ (θtω)h(t, x, u), χ 2 = du dt and u = u(t 0 ). Therefore, u is a weak solution of (16) supplement with initial and boundary condition (5)- (6).…”

Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains

“…The random attractors for stochastic partial equations and stochastic lattice dynamical systems have been widely discussed by many authors, see, e.g., [3,4,43,51] in the autonomous stochastic equations, and [25,28,46,49,48,50,53] in the non-autonomous case. In recent years, there are some results on random attractors for stochastic equations with the fractional Laplacian (−∆) α with α ∈ ( 1 2 , 1) and α ∈ (0, 1) in [26,27,29,31,32,33,42,44].…”