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

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...

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Bibliographic Details
Published in:	Mathematics (Basel) Vol. 9; no. 24; p. 3248
Main Authors:	Shi, Linfei, Cheng, Wenguang, Mao, Jinjin, Xu, Tianzhou
Format:	Journal Article
Language:	English
Published:	Basel MDPI AG 01-12-2021
Subjects:	blow-up Boundary conditions Boundary value problems caputo derivative Eigenvalues global existence Laplace transforms Mathematical functions Porous materials Reaction-diffusion equations reaction–diffusion equation
Online Access:	Get full text
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Summary:	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.
ISSN:	2227-7390 2227-7390
DOI:	10.3390/math9243248