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

Ao Zhang, Yanjie Zhang, Sanyang Zhai, Li Lin

Abstract

This article discusses the weak pullback attractors for a damped stochastic fractional Schrödinger equation on $\mathbb{R}^n$ with $n\geq 2$. 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 $|u|^{2σ}u$ 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 $L^ρ\left(Ω; L^2\left(\mathbb{R}^n\right)\right)$. This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations.

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 with . 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 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 . This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations.

Paper Structure

This paper contains 11 sections, 6 theorems, 37 equations.

Table of Contents

  1. Introduction.
  2. Statement of the main results
  3. Assumptions and preliminaries
  4. Main results
  5. Existence of global solution
  6. Radial local theory
  7. Proof of Theorem \ref{['theorem0']}
  8. Existence for global attractor
  9. Moment estimates
  10. Proof of Theorem \ref{['theorem2']}
  11. Acknowledgments

Key Result

Theorem 2.4

Let $n \geq 2$, $\alpha \in\left[\frac{n}{2 n-1}, 1\right)$, $0\leq\sigma<\frac{2 \alpha}{n-2 \alpha}$ and $\Phi\in L_{HS}\left(L^2(\mathbb{R}^n), L^2(\mathbb{R}^n)\right)$. Assume that $(r,p)=(\frac{4(\sigma+1)\alpha}{n\sigma}, 2\sigma+2)$, and that the radial initial data $u_0\in L^{\rho}\left(\Om and $u(\cdot,\omega)\in \mathcal{C}(0, T; L^2\left(\mathbb{R}^n)\right)$ for $\mathbb{P}$-a.a. $\om

Theorems & Definitions (14)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.2
  • Theorem 2.5
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • ...and 4 more