On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

Xiulan Wu; Yaxin Zhao; Xiaoxin Yang; Xiulan Wu; Yaxin Zhao; Xiaoxin Yang

doi:10.3934/cam.2024025

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 528-553. doi: 10.3934/cam.2024025

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On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

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

Received: 12 December 2023 Revised: 02 February 2024 Accepted: 10 July 2024 Published: 18 July 2024
35A02, 35B44, 35L67

In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.
- Non-Newton filtration equation,
- singular potential,
- logarithmic nonlinearity,
- global existence,
- decay,
- blow-up
Citation: Xiulan Wu, Yaxin Zhao, Xiaoxin Yang. On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity[J]. Communications in Analysis and Mechanics, 2024, 16(3): 528-553. doi: 10.3934/cam.2024025

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Abstract

In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.

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