Blowup and MLUH stability of time-space fractional reaction-diffusion equations

Peng Gao; Pengyu Chen; Peng Gao; Pengyu Chen

doi:10.3934/era.2022170

Electronic Research Archive

2022, Volume 30, Issue 9: 3351-3361. doi: 10.3934/era.2022170

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Blowup and MLUH stability of time-space fractional reaction-diffusion equations

Peng Gao ,
Pengyu Chen ^,

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Academic Editor: Arran Fernandez

Received: 01 May 2022 Revised: 02 July 2022 Accepted: 05 July 2022 Published: 14 July 2022

In this paper, we consider a class of nonlinear time-space fractional reaction-diffusion equations by transforming the time-space fractional reaction-diffusion equations into an abstract evolution equations in a fractional Sobolev space. Based on operator semigroup theory, the local uniqueness of mild solutions to the reaction-diffusion equations is obtained under the assumption that nonlinear function is locally Lipschitz continuous. On this basis, a blowup alternative result for unique saturated mild solutions is obtained. We further verify the Mittag-Leffler-Ulam-Hyers stability of the nonlinear time-space fractional reaction-diffusion equations.
- time-space fractional reaction-diffusion equations,
- Sobolev space,
- saturated mild solutions,
- local uniqueness,
- blowup alternative result
Citation: Peng Gao, Pengyu Chen. Blowup and MLUH stability of time-space fractional reaction-diffusion equations[J]. Electronic Research Archive, 2022, 30(9): 3351-3361. doi: 10.3934/era.2022170

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Abstract

In this paper, we consider a class of nonlinear time-space fractional reaction-diffusion equations by transforming the time-space fractional reaction-diffusion equations into an abstract evolution equations in a fractional Sobolev space. Based on operator semigroup theory, the local uniqueness of mild solutions to the reaction-diffusion equations is obtained under the assumption that nonlinear function is locally Lipschitz continuous. On this basis, a blowup alternative result for unique saturated mild solutions is obtained. We further verify the Mittag-Leffler-Ulam-Hyers stability of the nonlinear time-space fractional reaction-diffusion equations.

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