Blow-Up and Global Existence of Solutions for the Time Fractional Reaction–Diffusion Equation

Linfei Shi, Wenguang Cheng, Jinjin Mao and Tianzhou Xu
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Linfei Shi: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Wenguang Cheng: Department of Mathematics, Yuxi Normal University, Yuxi 653100, China
Jinjin Mao: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Tianzhou Xu: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Mathematics, 2021, vol. 9, issue 24, 1-9

Abstract: In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial data, we obtain two sufficient conditions for the blow-up of the solutions in finite time. Moreover, the existence of global solutions is studied when the initial data is small enough. Finally, the long-time behavior of bounded solutions is analyzed.

Keywords: caputo derivative; reaction–diffusion equation; blow-up; global existence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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