Liouville theorem for subcritical nonlinear heat equation

TL;DR

This paper addresses nonnegative ancient solutions to the subcritical semilinear heat equation $u_t=\Delta u+u^p$ in $\mathbb{R}^n\times(-\infty,0)$. It develops a Li-Yau-type gradient estimate, $u\,u_t \ge C_1|\nabla u|^2+C_2 u^{p+1}$, for $n\ge3$ and $1<p<\frac{n+2+\sqrt{n^2+8n}}{2(n-1)}$, with constants depending only on $n,p$, and extends this to settings on manifolds with nonnegative Ricci curvature and to Neumann problems. By pairing the Li-Yau-type inequality with Merle-Zaag's Liouville result, it achieves a Liouville-type theorem: a nonnegative ancient solution is either identically zero or takes a specific self-similar form $u(x,t)=\kappa(T_0-t)^{-1/(p-1)}$. The work also discusses extensions to broader geometric settings and notes implications for the rigidity of subcritical parabolic equations. Overall, it provides a gradient estimate-based route to Liouville-type classifications for subcritical nonlinear parabolic 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.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2: MZ98MZ00
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.3
• Proposition 2.1: Li-Yau type inequality
• proof
• Claim 2.1