Gradient estimates for the nonlinear parabolic equation with two exponents on Riemannian manifolds

TL;DR

This paper derives gradient estimates and Liouville theorems for positive solutions of a class of nonlinear parabolic equations with two exponents on complete noncompact Riemannian manifolds, extending previous results.

Contribution

It extends gradient estimates and Liouville theorems to a broader class of nonlinear parabolic equations with two exponents on Riemannian manifolds.

Findings

Established Souplet-Zhang type gradient estimates for positive solutions.

Proved Liouville theorems for positive ancient solutions.

Extended previous results to more general equations with two exponents.

Abstract

In this paper, we study the nonlinear parabolic equation with two exponents on complete noncompact Riemannian maniflods. The special types of such equation include the Fisher-KPP equation, the parabolic Allen-Cahn equation and the Newell-Whitehead equation. We get the Souplet-Zhang's gradeint estimates for the positive solutions to such equation. We also obtain the Liouville theorem for positive ancient solutions. Our results extend those of Souplet-Zhang (Bull. London. Math. Soc. 38:1045-1053, 2006) and Zhu (Acta Mathematica Scientia 36B(2): 514-526, 2016).