Harnack inequalities for a class of heat flows with nonlinear reaction terms

Abstract

A class of semilinear heat flows with general nonlinear reaction terms is considered on complete Riemannian manifolds with Ricci curvature bounded from below. Two types of (space-time and space only) gradient estimates are established for positive solutions to the flow, and the corresponding Harnack inequalities are obtained to allow for comparison of solutions. Some specific examples of the reaction term such as logarithmic reaction, Fisher-KPP and Allen-Cahn equations are discussed as applications of the estimates so derived. Referring to logarithmic nonlinearities, some discussions are made on Liouville type properties of bounded solutions.

Introduction

The main aim of this paper is to establish Li-Yau type (space-time) and Hamilton-Souplet-Zhang type (space only) Harnack inequalities for positive solutions $w(x,t)$ to a class of heat flows with nonlinear reactions

$$\frac{\partial }{\partial t}w(x,t)=\mathrm{\Delta }w(x,t)+g(w(x,t))$$

on an n-dimensional complete (compact or noncompact) Riemannian manifold M, where $t\in [0,\infty )$ and $g(\cdot )$ is a sufficiently smooth nonlinear function on $(0,\infty )$ and $0<w=w(x,t)$ is assumed smooth, atleast $C^2$ in x-variable and $C^1$ in t-variable. Here, $\frac{\partial w}{\partial t}$, which may sometimes be written as $w_t$, is the time derivative of w and Δ is the Laplace-Beltrami operator, while the term Δw accounts for the diffusion of heat. Thus (1.1) can be viewed as a reaction-diffusion model with ubiquitous applications in sciences and engineering. If w is a stationary solution, that is $w_t=0$, then (1.1) becomes a generalised Poisson equation. The nonlinear reaction term appearing in (1.1) includes several cases from biology, physics and geometry. For examples

• The case $g(w)=cw(1-w),c>0$, $0<w\le 1$, is the Fisher-KPP equation (see [24] in one dimension and [33] in two dimension), where $w(x,t)$ is the population density of genes at time t and $w(1-w)$ is known as logistic growth function [7], [34]. In this case, (1.1) is related to Fitzhugh-Nagumo equation [25] which is used to model impulse growth in nerve axons. It is also applicable in the studies of flame propagation and nuclear reactors, combustion theory and heat transfer, [36], [47].

• The case $g(w)=cw(1-w^2),c>0$, $0<w\le 1$, is the Allen-Cahn equation used in the theory of phase separation in iron alloys, including order-disorder transitions [8]. Allen-Cahn equation is well studied in literature (see [14], [21]), while its connection with the theory of minimal hypersurfaces in differential geometry has been greatly exploited by several authors, see [18], [23], [32] and references therein for instance. It also gave rise to the popular De Giorgi conjecture [20] which states that an entire solution to the steady state part with $|w|\le 1$ in ${\mathbb{R}}^n$, $n\le 8$, which is monotone in one direction is necessarily one dimensional (see [28] and [9] for the case $n=2$ and $n=3$, respectively).

• The case $g(w)=aw{(\log w)}^{\alpha }$, $a,\alpha \in \mathbb{R}$, is the evolution equation with logarithmic nonlinearity (see [44] for instance). When $\alpha =1$, the stationary part of the equation can be linked with gradient Ricci solitons and logarithmic Sobolev inequalities [22], [27]. Recall that the gradient Ricci solitons are self similar solutions to the Ricci flow, while the logarithmic Sobolev inequalities are infinite dimensional version of Sobolev inequalities widely applied in the theory of partial differential equations and constructive quantum field theory. The logarithmic Schrödinger equation

  $$i{u}_t=(\epsilon \mathrm{\Delta }-V)u+u\log |u{|}^{2}on{\mathbb{R}}^n,$$

  where V is a certain potential function, has considerable application in quantum mechanics, quantum optics, theory of superfluidity, effective quantum gravity (see [51] and the reference therein).

• The case $g(w)=|w{|}^{b-1}w,b>1$, is also of physical and geometric interests, we refer to [16], [17], [42], [43] and the references therein. Precisely, the authors in [43] studied the entire solution (see [42] for the steady state) in the case $1<b<p_{\star }$, where $p_{\star }=\frac{n+2}{n-2}$ for $n\ge 3$ and $p_{\star }=\infty$ for $n=1,2$, is the critical Sobolev exponent, and they obtained Liouville type theorem for radial solutions. The cases $b=2$ and $1<b<\frac{n(n+2)}{{(n-1)}^2}$ were respectively studied in [16] and [17] for ancient solutions on Riemanniqan manifolds.

• The nonlocal heat flow preserving $L^2$-norm on a closed manifold is another important case for

  $$g(w)=\lambda (t)w+A(x,t)$$

  with $\lambda (t):=-{\int }_M(w\mathrm{\Delta }w+wA)dv$ and ${\int }_M{w}^{2}dv=1$ chosen to preserve the norm. This case has diverse interpretations as well [12], [38].

• Consider the case $g(w)=a{w}^p-b{w}^q$, $a,b>0$, $q>p\ge 1$. The model (1.1) (with $g(w)=a{w}^p-b{w}^q$) covers Yamabe-type equation [1], [40], Lichnerowich-type equation [22] and some other parabolic equations of mathematical physics values.

In the seminal paper [37], Li and Yau considered Schrödinger type heat flow

$$\frac{\partial u}{\partial t}=\mathrm{\Delta }u+q(x,t)u$$

on a general manifold, where $q(x,t)$ is restricted to satisfying $C^2$ in x and $C^1$ in t. Precisely, they arrived at the following gradient estimates for $q(x,t)\equiv 0$

$$\frac{|\mathrm{\nabla }u{|}^2}{u^2}-\beta \frac{u_t}{u}\le \frac{n{\beta }^{2}k}{2(\beta -1)}+\frac{n{\beta }^2}{2t},\beta >1$$

on a Riemannian manifold whose Ricci curvature, $Ric\ge -k,k\ge 0$. They then applied (1.2) to deducing the following Harnack inequality

$$u(x_1,t_1)\le u(x_2,t_2){\left(\frac{t_2}{t_1}\right)}^{\frac{n\beta }{2}}{e}^{\mathrm{\Gamma }},$$

where $\mathrm{\Gamma }:=\frac{\beta d^2(x_1,x_2)}{4(t_2-t_1)}+\frac{\beta nk}{2(\beta -1)}(t_2-t_1)$, $d(x_1,x_2)$ is the geodesic distance between points $x_1$ and $x_2$ in M, $0<t_1<t_2$ and ∇ is the gradient operator. The above Harnack inequality (1.3) was in turn used to prove various bounds on the fundamental solution (heat kernel) to the linear heat equation. Li-Yau gradient estimates can also be used to derive Liouville type theorems for parabolic equations, estimates on Green's functions and various bounds on Laplacian eigenvalues.

The Harnack inequalities of type (1.3) can only be used to compare solutions at different time, since the gradient type estimates of the form (1.2) are time dependent. In order to circumvent this shortcoming, elliptic type (space only) gradient estimates were developed, first by Hamilton [30] on closed manifolds, and later by Souplet and Zhang [46] on complete noncompact manifolds, while their associated Harnack inequalities read as

$$u(x_1,t)\le u{(x_2,t)}^{\mathcal{A}(d(x_1,x_2),t)}{e}^{1-\mathcal{A}(d(x_1,x_2),t)},$$

which can thus be used to compare solutions at the same time.

Since the appearance of the above results, gradient and Harnack estimates have turned out to become one of the most powerful tools in geometric analysis and the theory of linear and nonlinear partial differential equations. Li-Yau Harnack inequalities played vital role in Perelman's proof of noncollapsing theorem for the Ricci flow [41], which eventually led to the final resolution of Poincaré conjecture. Quite a large number of literature has considered this topic under various settings. For instance, the authors [10], [50] found appropriate space-time and space only gradient estimates on bounded solution to the heat equation on domains evolving by the Ricci flow. In a similar vein, the first author [2] derived Harnack inequalities via Perelman type entropy formula under the Ricci-harmonic flow and in [3], [4] derived Li-Yau gradient estimates on heat type equation with potential under general geometric flow. Ma and Cheng [38] extended Li-Yau gradient estimates to $L^2$-norm preserving nonlocal heat flows on closed manifolds. Wang [48], on the other hand, extended Ma and Cheng's result to the Ricci flow setting. The first author [5], Bǎileşteanu [11] and Hou [31] studied differential Harnack inequalities and gradient estimates in relation to the bounded solutions of Allen-Cahn equation. Differential Harnack estimates and gradient estimates were also considered for Fisher-KPP equation in [15] and [26]. Wu [49] considered space-only time gradient estimates on weighted heat equation, the first author [1] considered nonlinear weighted parabolic equation of Yamabe type and in [6] weighted heat equation with logarithmic nonlinearity, Chen and Zhao [19] and Ma and Zeng [39] studied this concept with respect to certain nonlinear parabolic equations (see also the recent paper by Dung and Khanh [22]). We refer to Li and Xu [35] for a comprehensive discussion on the development of Harnack inequalities for linear heat equation on manifolds and their applications to Perelman type entropies, Liouville type results and heat kernel estimates.

In this paper, we are concerned with space-time and space only gradient estimates and their associated Harnack inequalities for the heat flow with nonlinear reaction term (1.1). The motivation for this stems out of the geometric and physical applications of such a class of equations as highlighted above. The results of this paper therefore provide a unified treatment of a large class of reaction-diffusion equations with nonlinear reaction terms. The rest of the paper is structured as follows. Section 2 is devoted to discussing space-time gradient estimates and several Li-Yau Harnack type inequalities are derived as corollaries. Specific estimates are shown for some semilinear heat equations in this class. In Section 3 we present space only gradient estimates and their corresponding Harnack inequalities for bounded positive solutions. Some examples are given to complement the results.