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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Abstract

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
Volume27
Issue number3
Early online date6 May 2020
DOIs
Publication statusPublished - 30 Jun 2020

Keywords

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

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    Asogwa, S. A., Foondun, M., Mijena, J. B., & Nane, E. (2020). Critical parameters for reaction-diffusion equations involving space-time fractional derivatives. Nonlinear Differential Equations and Applications NoDEA, 27(3), [30]. https://doi.org/10.1007/s00030-020-00629-9