Differential Harnack estimates for Fisher’s equation

Abstract

In this paper, we derive several differential Harnack estimates (also known as Li-Yau-Hamilton-type estimates) for positive solutions of Fisher's equation. We use the estimates to obtain lower bounds on the speed of traveling wave solutions and to construct classical Harnack inequalities.Date: October 7, 2016. 1 2 XIAODONG CAO, BOWEI LIU, IAN PENDLETON, AND ABIGAIL WARD Our work introduces and proves three Li-Yau-Hamilton-type Harnack inequalities which constrain positive functions satisfying the Fisher-KPP equation on an arbitrary Riemannian manifold M n . Depending on the setting we obtain different inequalities. The study of differential Harnack inequalities was first initiated by P. Li and S.-T. Yau in [10] (also see [1]). Harnack inequalities have since played an important role in the study of geometric analysis and geometric flows (for example, see [7,13]). Applications have also been found to the study of nonlinear parabolic equations, e.g. in [8]. One of these is a recent reproof of the classical result of H. Fujita [6], which states that any positive solution to the Endangered Species Equation in dimension n,blows up in finite time provided 0 < n(p − 1) < 2; see [3].When the dimension falls into a certain range we can integrate our differential Harnack inequality along any space-time curve to obtain a classical Harnack inequality which allows us to compare the values of positive solutions at any two points in space-time when time is large.The organization for the paper is as follows: In Section 2 we present the precise formulations and the proofs of our two inequalities governing closed manifolds. In Section 3 we state and prove a similar Harnack inequality for complete noncompact manifolds. In Section 4, we end the paper with the aforementioned results on the minimum speed of traveling wave solutions and classical Harnack inequalities.

Gradient estimates for a nonlinear parabolic equation and Liouville theorems

“…If f is constant, p(x, t) and q(x, t) are identically zero, then the theorem returns to the well-known Li-Yau gradient estimate [29]. More parabolic gradient estimates for special cases of the equation (1.1) were proved in [11,13,27,30].…”

“…In 2006, Souplet and Zhang [38] proved a local elliptic form by adding a logarithmic correction term. Recently, many authors extended the Li-Yau and Hamilton-Souplet-Zhang gradient estimates to the other heat-type equations; see for example [3,11,12,13,18,42,43,44,50,51] and references therein.…”

A Differential Harnack Inequality for the Newell-Whitehead-Segel Equation

“…By Lemma 4 of [3], we know that in the constant form this is a valid solution to this differential equation. The only other behavior we desire is that ϕ(t) > 0 for all time and that ϕ(t) diverges towards positive infinity as t → 0, so that we can ensure that H(x, t) starts off positive and therefore its first zero must be a negative time derivative.…”

Gradient Estimates for the Nonlinear Parabolic Equation with Two Exponents on Riemannian Manifolds

“…It describes the propagation of an evolutionarily advantageous gene in a population and has many applications. Cao, Liu, Pendleton and Ward [4] derived some differential Harnack estimates for positive solutions to (1.2) on Riemannian manifolds. Geng and the author [7] extended the result of [4].…”