Blow-up results for one dimensional Caputo fractional reaction diffusion equation

Abstract

We study the blow up problems for ordinary Caputo fractional differential equation and the time dependent Caputo fractional reaction diffusion equation in one dimensional space. We establish that the solution of the differential equation of the integer order which blows up in finite time can be used as a tool to construct a lower solution to the equation of the fractional order, under suitable conditions. Hence, we obtain the blow up of the solution of the differential equation of integer order implies that the blow up of the solution of the differential equation of fractional order. For that purpose, we use the known comparison results of Caputo ordinary fractional equation and Caputo fractional reaction diffusion equation. We also prove the blow up in finite time of the Caputo fractional reaction diffusion equation using a similar method which has been used to prove the blow-up of the solution of ordinary reaction diffusion equation.