Harnack Estimate for the Endangered Species Equation

TL;DR

This paper establishes a differential Harnack inequality for the Endangered Species Equation, a nonlinear parabolic PDE, leading to insights about solution blowup and addressing a question posed by Hamilton in 2011.

Contribution

It introduces a novel differential Harnack inequality for the Endangered Species Equation and applies it to analyze solution blowup and answer Hamilton's question.

Findings

Positive solutions blow up in finite time.

The Harnack inequality provides new bounds for solutions.

Partially answers Hamilton's 2011 question.

Abstract

We prove a differential Harnack inequality for the Endangered Species Equation, a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an application of this inequality, we will show that positive solutions to this equation must blowup in finite time. We also use this inequality to partially answer a question of Hamilton, 2011.