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
DOI
10.1007/s00030-020-00629-9
Recommended Citation
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, 27(3).
Comments
© 2020, The Author(s).