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

2024/08/06 by Liang, Wenguo, Zhang, Zhengce
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.02893

Abstract

This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation u_t-Δu=u^p+M|∇ u|^q in Ω× I⊂ \R^N× \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-Véron, Garcia-Huidobro and Véron (2020) \citeveron-sum) to the parabolic case.

Related

• Optimal Liouville theorems for superlinear parabolic problems
• Liouville theorem for subcritical nonlinear heat equation
• An optimal Liouville theorem for the linear heat equation with a\n nonlinear boundary condition
• A Liouville property for eternal solutions to a supercritical semilinear\n heat equation
• Pointwise Bounds and Blow-up for Systems of Semilinear Parabolic Inequalities and Nonlinear Heat Potential Estimates