Dynamics of non-autonomous fractional reaction-diffusion equations on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> driven by multiplicative noise

Kaixuan Zhu; Ji Li; Yongqin Xie; Mingji Zhang

doi:10.3934/dcdsb.2020376

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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 and Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376

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References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666. doi: 10.3934/dcdsb.2013.18.643.
[3]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.
[4]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.
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[11]	T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047.
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[14]	T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. B, 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525.
[15]	D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.
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[18]	I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. doi: 10.1080/1468936042000207792.
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[28]	H. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186. doi: 10.1016/j.jde.2018.07.028.
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[32]	A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137. doi: 10.1016/j.jde.2018.02.011.
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[36]	X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 1–9. doi: 10.1063/1.2235026.
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[45]	H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198. doi: 10.1016/j.na.2015.06.033.
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American Institute of Mathematical Sciences

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

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

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