Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity

Caihong Gu; Yanbin Tang; Caihong Gu; Yanbin Tang

doi:10.3934/nhm.2023005

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 109-139. doi: 10.3934/nhm.2023005

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Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity

Caihong Gu ¹,
Yanbin Tang ^{1,2
,
,}

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China
2.
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

Received: 21 September 2022 Revised: 29 October 2022 Accepted: 02 November 2022 Published: 09 November 2022

In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.
- Drift-diffusion system,
- Keller-Segel equations,
- global solutions,
- fractional Laplacian,
- asymptotic behavior,
- multi-linear operator
Citation: Caihong Gu, Yanbin Tang. Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity[J]. Networks and Heterogeneous Media, 2023, 18(1): 109-139. doi: 10.3934/nhm.2023005

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Abstract

In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.

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