Differential Harnack Estimates for a Semilinear Parabolic System

Abstract

In this paper, we prove differential Harnack inequalities for positive solutions of a semilinear parabolic system on hyperbolic space. We use the inequalities to construct classical Harnack estimates by integrating along space-time.

1. Introduction

In this paper, we study the following problem:where are positive constants.

P. Li and S.-T.Yau in [1] were first pioneers to the study of differential Harnack inequalities which were brought to general parabolic geometric flows by R. Hamilton (see [2]). Using these inequalities can derive ancient solutions, bounds on gradient Ricci solitons, Holder continuity. Differential Harnack inequalities are important aspects of properties of partial differential equations. Paper [3] described differential Harnack inequalities to the initial value problem of a semilinear parabolic equation when the semilinear term is . There have been numerous interesting results on the properties of solutions of partial differential equations, such as existence of solutions [4–17], nonexistence and blow-up of solutions [18–22], and asymptotic behaviors of solutions [23–28].

Let be positive smooth solutions to (1) and . The main object of our study is the following Harnack quantities:where , , , and , will be chosen suitably .

We will derive our differential Harnack estimate.

Theorem 1. Let be positive classical solutions to (1), and . If , , , and satisfy then we havefor all .

The paper is organized as follows. In Section 2 we prove Theorem 1 which describes differential Harnack estimate. There are applications of Theorem 1 in Section 3.

2. Harnack Estimate

In this section, we shall first obtain our differential Harnack inequalities, relying on the parabolic maximum principle.

Lemma 2. Suppose are positive solutions to (1) and and are defined as in (2). Assume that , , , and satisfy Then we have where

Proof. Substituting into (1), we have Using the above equations, we have Furthermore, applying (2) and Cauchy-Schwarz inequality yields If , the above inequality is First note the following inequality:If and , the above inequality isFor , we have similar results. If , the above inequality is If , the above inequality is This completes the proof of Lemma 2.

Next, we need to compute specific and that guarantee for the maximum principle to be applicable where .

Lemma 3. Assume , , , and If , , for , then for some , and .

Proof. From , and , , we get and .
Now applying Lemma 2 yields By the definitions of and , we obtain and If then and .
This proves Lemma 3.

Proof of Theorem 1. Choose for . Note thatAssume that there exists a first time and point where and . At ; we haveLemma 3 implies thatThis is a contradiction. Assume that there exist a first time and point where and . At , we haveLemma 3 implies thatThis is a contradiction. Furthermore, and cause the same contradiction as and . Thus and for all .
Taking which obtains then gives the desired result.

3. Applications

In this section, we shall give an application of Theorem 1. We integrate along space-time to derive a classical Harnack inequality.

3.1. Classical Harnack Inequality

In this subsection, we integrate our differential Harnack inequality of Theorem 1 along space-time to derive a classical Harnack inequality.

Proposition 4. Let be positive classical solutions to (1) and . Suppose that , and . Assume further that , and , . Then we have

Proof. Define the one-variable functions asfor any path such that , .
Applying Theorem 1, we haveIt yields that where , , and . Similarly,for , , and .
Byapplying gives Proposition 4.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (No. 11801306).