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

• Published in: AIMS mathematics Vol. 7; no. 7; pp. 12913 - 12934
• Main Author: Li, Zhiqiang
• Format: Journal Article
• Language: English
• Published: AIMS Press 01-01-2022
• Subjects: caputo-hadamard derivative; finite time blow-up; fixed point argument; fractional laplacian; nonlinear memory
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2022715

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.