A Differential Harnack Inequality for the Newell-Whitehead Equation

TL;DR

This paper establishes a differential Harnack inequality for positive solutions to the Newell-Whitehead equation, enabling new insights into solution properties, classical inequalities, and solution types like standing and traveling waves.

Contribution

It introduces a Li-Yau-Hamilton type differential Harnack estimate specifically for the Newell-Whitehead equation, a novel analytical tool for this PDE.

Findings

Derived a classical Harnack inequality for the equation

Characterized standing solutions and traveling wave solutions

Provided new estimates for positive solutions

Abstract

This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including deriving a classical Harnack inequality, and characterizing standing solutions and traveling wave solutions.