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

Sunday A. Asogwa, Mohammud Foondun, Jebessa B. Mijena, Erkan Nane

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6 Citations (Scopus)
11 Downloads (Pure)


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 < η⩽ η c, there is no global solution other than the trivial one while for η> η c, non-trivial global solutions do exist. The critical parameter η c is shown to be 1η∗ where η∗:=supa>0{supt∈(0,∞),x∈Rdta∫RdG(t,x-y)V0(y)dy<∞}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.

Original languageEnglish
Article number30
Number of pages22
JournalNonlinear Differential Equations and Applications NoDEA
Issue number3
Early online date6 May 2020
Publication statusPublished - 30 Jun 2020


  • space time fractional partial differential equations
  • Fujita type blow-up conditions
  • critical exponents
  • Caputo derivatives
  • Dirichlet boundary condition


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