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

Asogwa, Sunday A. and Foondun, Mohammud and Mijena, Jebessa B. and Nane, Erkan (2020) Critical parameters for reaction-diffusion equations involving space-time fractional derivatives. Nonlinear Differential Equations and Applications NoDEA, 27 (3). 30. ISSN 1021-9722

[img]
Preview
Text (Asogwa-etal-NDEA-2020-Critical-parameters-for-reaction-diffusion-equations)
Asogwa_etal_NDEA_2020_Critical_parameters_for_reaction_diffusion_equations.pdf
Final Published Version
License: Creative Commons Attribution 4.0 logo

Download (423kB)| Preview

    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.