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Well-posedness and dynamics of fractional Fitz Hugh-Nagumo systems on R^(N) driven by nonlinear noise

Science China Mathematics 2021年第11期64卷 2395-2436页

作者： Renhai Wang Boling Guo Bixiang Wang Institute of Applied Physics and Computational Mathematics Beijing 100088China Department of Mathematics New Mexico Institute of Mining and TechnologySocorroNM 87801USA

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 Fitz Hugh-Nagumo systems driven by nonlinear noise defined on the entire space *** 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 *** mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner *** existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion *** 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 RNas well as the lack of smoothing effect on one component of the *** results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

关键词： fractional Fitz Hugh-Nagumo system weak pullback mean random attractor invariant measure nonlinear noise unbounded domain

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

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Science China Mathematics 2020年

作者： RenhaiWang BolingGuo BixiangWang InstituteofAppliedPhysicsandComputationalMathematics DepartmentofMathematics NewMexicoInstituteofMiningandTechnology

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 space RN. 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 RNas 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.

关键词： fractional FitzHugh-Nagumo system weak pullback mean random attractor invariant measure nonlinear noise unbounded domain

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