Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

Yaxin Zhao; Xiulan Wu; Yaxin Zhao; Xiulan Wu

doi:10.3934/math.20241113

AIMS Mathematics

2024, Volume 9, Issue 8: 22883-22909. doi: 10.3934/math.20241113

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Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

Yaxin Zhao ,
Xiulan Wu ^,

School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130000, China

Received: 30 May 2024 Revised: 07 July 2024 Accepted: 12 July 2024 Published: 24 July 2024
MSC : 35A01, 35B40, 35B44, 35K35

This paper considers a class of higher-order nonlocal parabolic equations with special coefficient. We apply the Faedo-Galerkin approximation method and cut-off technique to obtain the local solvability. Furthermore, based on the framework of the modified potential well, we get the global existence, asymptotic behavior, and blow-up of the weak solutions by the Hardy-Sobolev inequality when the initial energy is subcritical $ J(u_0) < d $. In the critical case of $ J(u_0) = d $, the above results have also been obtained. Finally, we utilize some new processing methods to gain the blow-up criterion in finite time with supercritical initial energy $ J(u_0) > 0 $.
- logarithmic nonlocal source,
- special diffusion process,
- global existence,
- asymptotic behavior,
- blow-up
Citation: Yaxin Zhao, Xiulan Wu. Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process[J]. AIMS Mathematics, 2024, 9(8): 22883-22909. doi: 10.3934/math.20241113

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Abstract

This paper considers a class of higher-order nonlocal parabolic equations with special coefficient. We apply the Faedo-Galerkin approximation method and cut-off technique to obtain the local solvability. Furthermore, based on the framework of the modified potential well, we get the global existence, asymptotic behavior, and blow-up of the weak solutions by the Hardy-Sobolev inequality when the initial energy is subcritical $ J(u_0) < d $. In the critical case of $ J(u_0) = d $, the above results have also been obtained. Finally, we utilize some new processing methods to gain the blow-up criterion in finite time with supercritical initial energy $ J(u_0) > 0 $.

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