Weak pullback attractors for damped stochastic fractional Schrödinger equation on $\mathbb{R}^n

Abstract

This article discusses the weak pullback attractors for a damped stochastic fractional Schrödinger equation on ${\mathbb{R}}^n$ with $n\ge 2$. By utilizing the stochastic Strichartz estimates and a stopping time technique argument, the existence and uniqueness of a global solution for the systems with the nonlinear term $|u{|}^{2\ddot{I}ƒ}u$ are proven. Furthermore, we define a mean random dynamical system due to the uniqueness of the solution, which has a unique weak pullback mean random attractor in $L^{\ddot{I}}\left(\hat{I}\text{c◯};L^2\left({\mathbb{R}}^n\right)\right)$. This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations.