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

Abstract

AbstractWe will look at reaction–diffusion type equations of the following type, $$\begin{aligned} \partial ^\beta _tV(t,x)=-(-\Delta )^{\alpha /2} V(t,x)+I^{1-\beta }_t[V(t,x)^{1+\eta }]. \end{aligned}$$ ∂ t β V ( t , x ) = - ( - Δ ) α / 2 V ( t , x ) + I t 1 - 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 $$0<\eta \leqslant \eta _c$$ 0 < η ⩽ η c , there is no global solution other than the trivial one while for $$\eta >\eta _c$$ η > η c , non-trivial global solutions do exist. The critical parameter $$\eta _c$$ η c is shown to be $$\frac{1}{\eta ^*}$$ 1 η ∗ where $$\begin{aligned} \eta ^*:=\sup _{a>0}\left\{ \sup _{t\in (0,\,\infty ),x\in \mathbb {R}^d}t^a\int _{\mathbb {R}^d}G(t,\,x-y)V_0(y)\,\mathrm{d}y<\infty \right\} \end{aligned}$$ η ∗ : = sup a > 0 sup t ∈ ( 0 , ∞ ) , x ∈ R d t a ∫ R d G ( t , x - y ) V 0 ( y ) d y < ∞ and $$G(t,\,x)$$ G ( t , x ) is the heat kernel of the corresponding unforced operator. $$V_0$$ V 0 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.