American Institute of Mathematical Sciences

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October  2021, 26(10): 5681-5705. doi: 10.3934/dcdsb.2020376

Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise

Kaixuan Zhu 1, , Ji Li 2, , Yongqin Xie 3, and  Mingji Zhang 4,,

1. 

Hunan Province Cooperative Innovation Center for the Construction, and Development of Dongting Lake Ecological Economic Zone & College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, 415000, China
2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
3. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China
4. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Mingji Zhang

Received  June 2020 Published  October 2021 Early access  December 2020

We examine the asymptotic behavior of the non-autonomous non-local fractional stochastic reaction-diffusion equations on $ \mathbb{R}^{N} $ with the nonlinearity $ f $ satisfying the polynomial growth of arbitrary order $ p-1 $ $ (p\geq2) $. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time. Then, we prove that the solution process is continuous from $ L^{2}(\mathbb{R}^{N}) $ to $ H^{s}(\mathbb{R}^{N}) $ with respect to initial data for $ s\in(0, 1) $. As an application of the higher-order integrability and the continuity, we obtain the pullback $ \mathcal{D} $-random attractors in $ L^{p}(\mathbb{R}^{N}) $ and $ H^{s}(\mathbb{R}^{N}) $, respectively. This is a natural and necessary extension of current existing results to further understand the dynamics of the underlying problem.

Keywords: Fractional stochastic reaction-diffusion equations, higher-order integrability, continuity, pullback random attractors.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35K57.
Citation: Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376
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