Liouville theorem for subcritical nonlinear heat equation

TL;DR

This paper establishes a Liouville theorem for nonnegative ancient solutions to a subcritical nonlinear heat equation by deriving a Li-Yau-type estimate and combining it with existing results.

Contribution

It introduces a Li-Yau-type estimate for the subcritical semilinear heat equation and proves a Liouville theorem for its solutions, extending previous understanding.

Findings

Derived a Li-Yau-type estimate for solutions.

Proved a Liouville theorem for the equation.

Extended Liouville theorems to subcritical nonlinear heat equations.

Abstract

We obtain a Li-Yau-type estimate for nonnegative ancient solutions to the subcritical semilinear heat equation $\frac{\p u}{\p t}=\De u+u^p$ in $\rz^n\times(-\infty,0)$. Then, we combine the Li-Yau type estimate and Melre-Zaag's result to prove the Liouville theorem of this equation.