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Mean random attractors of stochastic lattice fractional delay Gray–Scott equations in higher moment product sequence spaces

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Abstract

This paper is devoted to the study of mean attractors in some higher moment product sequence spaces for a three-component reversible lattice stochastic Gray–Scott equation, where the nonlinear terms have polynomial growth of arbitrary orders, and the diffusion term is a locally Lipschitz function with a time-delay effect. We first formulate the equation into an abstract one in \(\mathbb {L}^2=:\ell ^2\times \ell ^2\times \ell ^2\). We then establish the well-posedness and Itô’s energy equalities for the system in \(L^{2k}(\Omega , \mathcal {F}; \mathbb {L}^2)\times L^{2k}\big (\Omega , \mathcal {F}; L^{2k}((-\rho , 0), \mathbb {L}^2)\big )\) over the complete filtered probability space \((\Omega ,\mathcal {F}, \{\mathcal {F}_t\}_{t\in \mathbb {R}},\mathbb {P})\), where the index \(k\ge 1\) is arbitrary and \(\rho \ge 0\) is the time-delay parameter. Finally, we prove that the mean random dynamical system admits a unique mean random attractor in this high-order Banach space in the sense of Wang (J Dyn Differ Equ 31:2177–2204, 2019). Our results are new even in the product Hilbert space \(L^{2}(\Omega , \mathcal {F}; \mathbb {L}^2)\times L^{2}\big (\Omega , \mathcal {F}; L^{2}((-\rho , 0), \mathbb {L}^2)\big )\) when \(k=1\).

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Acknowledgements

Lianbing She was supported by the Science and Technology Foundation of Guizhou Province ([2020]1Y007) and School Level Foundation of Liupanshui Normal University (LPSSYKJTD201907). Renhai Wang was supported by the National Natural Science Foundation of China (12301299), the research fund of Qianshixinmiao [2022]B16, Natural Science Research Project of Guizhou Provincial Department of Education (no. QJJ[2023]011), and the research fund of Qiankehepingtairencai-YSZ[2022]022.

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  1. School of Mathematical Science, Guizhou Normal University, Guiyang, 550025, China

    Xiaolan Qin & Renhai Wang

  2. School of Mathematics and Statistics, Liupanshui Normal University, Liupanshui, 553004, China

    Lianbing She

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  1. Xiaolan Qin
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Communicated by Ti-Jun Xiao.

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Qin, X., She, L. & Wang, R. Mean random attractors of stochastic lattice fractional delay Gray–Scott equations in higher moment product sequence spaces. Banach J. Math. Anal. 17, 82 (2023). https://doi.org/10.1007/s43037-023-00310-0

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