Critical parameters for reaction-diffusion equations involving space-time fractional derivatives
We will look at reaction-diffusion type equations of the following type, $$\partial^β_tV(t,x)=-(-Δ)^{α/2} V(t,x)+I^{1-β}_t[V(t,x)^{1+η}].$$ 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 $0<η\leqη_c$, there is no global solution other than the trivial one while for $η>η_c$, 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.
math.AP↗