Blow up of fractional reaction-diffusion systems with and without convection terms

Abstract

Based on the study of blow up of a particular system of ordinary differential equations, we give a sufficient condition for blow up of positive mild solutions to the Cauchy problem of a fractional reaction-diffusion system, and, by a comparison between the transition densities of the semigroups generated by $\Delta _\alpha :=-(-\Delta )^{\alpha /2}$ and $\Delta _\alpha +b(x)\cdot \nabla $ for $1\lt \alpha \lt 2$, $d\geq 1$ and $b$ in the Kato class on $\mathbb {R}^d$, we prove that this condition is also sufficient for the blow up of a fractional diffusion-convection-reaction system.