Blow-up solutions to the Cauchy problem of a fractional reaction-diffusion system

In this paper, we study the blow-up property of positive mild solutions to the Cauchy problem of a system of fractional reaction-diffusion equations. For the fundamental solution $P(t,x)$ of the fractional heat operator $\partial_{t}+(-\triangle)^{\frac{\beta}{2}}$ defined on the whole space $R^{N}$ , due to the properties of $P(t,x)$ established by H Yosida and some estimates of $P(t,x)$ developed by L Caffarelli and A Figalli, we first use an iteration method to establish the estimates of lower bounds of positive mild solutions; then we obtain the unboundedness of solutions for large time. Finally we give a sufficient condition that the positive mild solution to a fractional reaction-diffusion system blows up in finite time.