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 language | English |
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Article number | 30 |
Number of pages | 22 |
Journal | Nonlinear Differential Equations and Applications NoDEA |
Volume | 27 |
Issue number | 3 |
Early online date | 6 May 2020 |
DOIs | |
Publication status | Published - 30 Jun 2020 |
Keywords
- space time fractional partial differential equations
- Fujita type blow-up conditions
- critical exponents
- Caputo derivatives
- Dirichlet boundary condition