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Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on ℝN driven by nonlinear noise

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Abstract

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire spaceℝRN. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ℝN as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

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Acknowledgements

The first author was supported by the China Scholarship Council (Grant No. 201806990064). This work was done when the first author visited the Department of Mathematics at the New Mexico Institute of Mining and Technology. The first author expresses his thanks to all the people there for their kind hospitality.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Renhai Wang & Boling Guo

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

    Bixiang Wang

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  1. Renhai Wang
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  2. Boling Guo
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  3. Bixiang Wang
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Correspondence to Renhai Wang.

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Wang, R., Guo, B. & Wang, B. Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on ℝN driven by nonlinear noise. Sci. China Math. 64, 2395–2436 (2021). https://doi.org/10.1007/s11425-019-1714-2

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