A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source

TL;DR

This paper establishes gradient estimates and Liouville-type theorems for nonnegative solutions of a semilinear heat equation with nonlinear gradient source, extending elliptic results to the parabolic setting and deriving universal a priori bounds.

Contribution

It introduces new gradient estimates and Liouville theorems for parabolic equations with nonlinear gradient terms, generalizing previous elliptic results to the time-dependent case.

Findings

Gradient estimates for subcritical, critical, and supercritical q

Liouville theorems for ancient and entire solutions

Universal a priori estimates for local solutions

Abstract

This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-\Delta u=u^p+M|\nabla u|^q$ in $\Omega\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the local pointwise gradient estimates when $q$ is subcritical, critical and supercritical with respect to $p$. With these estimates, we can prove the parabolic Liouville-type theorems for time-decreasing ancient solutions. Next, we use Gidas-Spruck type integral methods to prove the Liouville-type theorem for the entire solutions when $q$ is critical. Finally, as an application of the Liouville-type theorem, we use the doubling lemma to derive universal priori estimates for local solutions of parabolic equations with general nonlinearities. Our approach relies on a parabolic differential inequality containing a suitable auxiliary function rather than Keller-Osserman type inequality, which allows us to generalize and extend the partial results of the elliptic equation (Bidaut-V\'{e}ron, Garcia-Huidobro and V\'{e}ron (2020) \cite{veron-sum}) to the parabolic case.