## Abstract

We consider the semilinear heat equation ut = ∆u + u^{p} on R^{N}. Assuming that N ≥ 3 and p is greater than the Sobolev critical exponent (N + 2)/(N − 2), we examine entire solutions (classical solutions defined for all t ∈ R) and ancient solutions (classical solutions defined on (−∞, T ) for some T < ∞). We prove a new Liouville-type theorem saying that if p is greater than the Lepin exponent pL := 1 + 6/(N − 10) (pL = ∞ if N ≤ 10), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of radial symmetry; in other ranges of supercritical p it is known not to be valid even in the class of radial solutions. Our other results include classification theorems for nonstationary entire solutions (when they exist) and ancient solutions, as well as some applications in the theory of blowup of solutions.

Original language | English (US) |
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Pages (from-to) | 413-438 |

Number of pages | 26 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 41 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2021 |

### Bibliographical note

Funding Information:2020 Mathematics Subject Classification. 35K58, 35B08, 35B44, 35B05, 35B53. Key words and phrases. Semilinear heat equation, entire solutions, ancient solutions, Liouville theorems, blowup. The first author is supported in part by NSF grant DMS-1856491. The second author is supported in part by VEGA Grant 1/0347/18 and by the Slovak Research and Development Agency under the contracts No. APVV-14-0378 and APVV-18-0308. ∗ Corresponding author.

## Keywords

- Ancient solutions
- Blowup
- Entire solutions
- Liouville theorems
- Semilinear heat equation