### 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 |
---|---|

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

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## Cite this

*Nonlinear Differential Equations and Applications NoDEA*,

*27*(3), [30]. https://doi.org/10.1007/s00030-020-00629-9