Critical parameters for reaction-diffusion equations involving space-time fractional derivatives

Abstract

We will look at reaction-diffusion type equations of the following type, $${\partial }_t^{\beta }V(t,x)=-(-\Delta {)}^{\alpha /2}V(t,x)+I_t^{1-\beta }[V(t,x{)}^{1+\eta }].$$ We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $0<\eta \le {\eta }_c$, there is no global solution other than the trivial one while for $\eta >{\eta }_c$, non-trivial global solutions do exist. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the time derivative induces a significant change in the behaviour of the solution.