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

We will look at reaction–diffusion type equations of the following type, ∂tβV(t,x)=-(-Δ)α/2V(t,x)+It1-β[V(t,x)1+η].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 \u3c η⩽ ηc, there is no global solution other than the trivial one while for η\u3e ηc, non-trivial global solutions do exist. The critical parameter ηc is shown to be 1η∗ where η∗,=supa\u3e0{supt∈(0,∞),x∈Rdta∫RdG(t,x-y)V0(y)dy\u3c∞}and G(t,x) is the heat kernel of the corresponding unforced operator. V is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution