# Entire and ancient solutions of a supercritical semilinear heat equation

## TL;DR

This paper studies entire and ancient solutions of a supercritical semilinear heat equation, proving a Liouville-type theorem for positive bounded radial solutions and classifying nonstationary solutions.

## Contribution

It establishes a new Liouville-type theorem for positive bounded radial solutions when p exceeds the Lepin exponent, and classifies entire and ancient solutions in supercritical regimes.

## Key findings

- All positive bounded radial entire solutions are steady states for p > p_L.
- Classification of nonstationary entire solutions when they exist.
- Applications to blowup behavior of solutions.

## Abstract

We consider the semilinear heat equation $u_t=\Delta u+u^p$ on ${\mathbb R}^N$. Assuming that $N\ge 3$ and $p$ is greater than the Sobolev critical exponent $(N+2)/(N-2)$, we examine entire solutions (classical solutions defined for all $t\in {\mathbb R}$) and ancient solutions (classical solutions defined on $(-\infty,T)$ for some $T<\infty$). We prove a new Liouville-type theorem saying that if $p$ is greater than the Lepin exponent $p_L:=1+6/(N-10)$ ($p_L=\infty$ if $N\le 10$), 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.