Title

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

Document Type

Article

Publication Date

2020

Publication Title

Nonlinear Differential Equations and Applications

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⩽ ηc, there is no global solution other than the trivial one while for η> ηc, non-trivial global solutions do exist. The critical parameter ηc 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.

Department

Mathematics

Volume Number

27

Issue Number

3

Comments

© 2020, The Author(s).

DOI

10.1007/s00030-020-00629-9

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