## Abstract

We consider the scalar semilinear heat equation u_{t}−Δu=f(u), where f:[0,∞)→[0,∞) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in L^{q}(Ω) for all non-negative initial data u_{0}∈L^{q}(Ω), when Ω⊂R^{d} is a bounded domain with Dirichlet boundary conditions. For q∈(1,∞) this holds if and only if limsup_{s→∞}s^{−(1+2q/d)}f(s)<∞; and for q=1 if and only if ∫_{1}^{∞}s^{−(1+2/d)}F(s)ds<∞, where F(s)=sup_{1≤t≤s}f(t)/t. This shows for the first time that the model nonlinearity f(u)=u^{1+2q/d} is truly the ‘boundary case’ when q∈(1,∞), but that this is not true for q=1. The same characterisations hold for the equation posed on the whole space R^{d} provided that limsup_{s→0}f(s)/s<∞.

Original language | English |
---|---|

Pages (from-to) | 1519-1538 |

Number of pages | 20 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 33 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Nov 2016 |

## Keywords

- Dirichlet heat kernel
- Dirichlet problem
- Instantaneous blow-up
- Local existence
- Non-existence
- Semilinear heat equation