The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory

Zhiqiang Li; Zhiqiang Li

doi:10.3934/math.2022715

AIMS Mathematics

2022, Volume 7, Issue 7: 12913-12934. doi: 10.3934/math.2022715

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Research article

The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory

Zhiqiang Li ^,

Department of Mathematics, Luliang University, Lvliang, Shanxi 033001, China

Received: 10 March 2022 Revised: 19 April 2022 Accepted: 20 April 2022 Published: 06 May 2022
MSC : 26A33, 35R11, 35B44

In this article, we focus on the blow-up problem of solution to Caputo-Hadamard fractional diffusion equation with fractional Laplacian and nonlinear memory. By virtue of the fundamental solutions of the corresponding linear and nonhomogeneous equation, we introduce a mild solution of the given equation and prove the existence and uniqueness of local solution. Next, the concept of a weak solution is presented by the test function and the mild solution is demonstrated to be a weak solution. Finally, based on the contraction mapping principle, the finite time blow-up and global solution for the considered equation are shown and the Fujita critical exponent is determined. The finite time blow-up of solution is also confirmed by the results of numerical experiment.
- Caputo-Hadamard derivative,
- fractional Laplacian,
- nonlinear memory,
- finite time blow-up,
- fixed point argument
Citation: Zhiqiang Li. The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory[J]. AIMS Mathematics, 2022, 7(7): 12913-12934. doi: 10.3934/math.2022715

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Abstract

In this article, we focus on the blow-up problem of solution to Caputo-Hadamard fractional diffusion equation with fractional Laplacian and nonlinear memory. By virtue of the fundamental solutions of the corresponding linear and nonhomogeneous equation, we introduce a mild solution of the given equation and prove the existence and uniqueness of local solution. Next, the concept of a weak solution is presented by the test function and the mild solution is demonstrated to be a weak solution. Finally, based on the contraction mapping principle, the finite time blow-up and global solution for the considered equation are shown and the Fujita critical exponent is determined. The finite time blow-up of solution is also confirmed by the results of numerical experiment.

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