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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 289168, 10 pages
http://dx.doi.org/10.1155/2012/289168
Research Article

Uniqueness of Traveling Waves for a Two-Dimensional Bistable Periodic Lattice Dynamical System

Department of Applied Mathematics, National Chung Hsing University, 250 Kuo Kuang Road, Taichung 402, Taiwan

Received 28 December 2011; Accepted 30 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Chin-Chin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study traveling waves for a two-dimensional lattice dynamical system with bistable nonlinearity in periodic media. The existence and the monotonicity in time of traveling waves can be derived in the same way as the one-dimensional lattice case. In this paper, we derive the uniqueness of nonzero speed traveling waves by using the comparison principle and the sliding method.

1. Introduction

In this paper, we study the following two-dimensional (2D) lattice dynamical system: where is a function in and We assume that the coefficients are positive and bounded such that for some positive integer . Furthermore, we consider the case of bistable nonlinearity, namely, for some constant . For simplicity, we only consider the case when .

We are interested in (planar) traveling wave solutions of (1.1) such that for some (speed) and for any in the direction for some .

The study of lattice dynamical systems has attracted a lot of attention for past years. In particular, traveling wave solutions are important due to the wide applications of these special solutions. For example, the invading of one species to another can be described by traveling wave solutions (see, e.g., [1, 2]). The lattice dynamical system arises, for example, when the habitat is divided into discrete niches in certain biology models. We refer the reader to, for example, [39] for monostable nonlinearity and [1016] for bistable nonlinearity in a one-dimensional lattice. In particular, in [16] the authors studied a very general model with bistable nonlinearity in a 1D lattice. Our purpose of this paper is to extend the result of [16] to the case of multidimensional lattice. For the study of multidimensional lattice, we refer to [1719]. For the simplicity of presentation, we will only consider the 2D lattice dynamical system (1.1). Our results can be easily extended to the more general case with a convection term or spatially dependent nonlinearity as in [16].

In a similar manner as that in [16] for 1D lattice case, we can prove the existence of traveling wave solutions of (1.1)–(1.6) with profile and speed , by transforming the problem (1.1)–(1.6) into an integral formulation. Moreover, if the speed , then we can obtain for all and . We will not repeat the proof here and focus on the study of the uniqueness of nonzero speed traveling waves. The uniqueness is in the sense that if there exist two traveling waves with nonzero speeds, then these two speeds are the same, and two wave profiles are the same except a translation. Due to that the nonlinearity is independent of spatial variable, our proof of the uniqueness is simpler and more transparent than that in [16]. In fact, motivated by the work of Fife and McLeod [20], Lemma 3.1 (below) provides some estimations in terms of a given traveling wave solution for the solution to the initial value problem for (1.1) with certain initial condition. Moreover, with Lemma 3.1, we employ the idea of moving coordinate and a sliding method to complete the proof of uniqueness (see Theorem 3.3).

This paper is organized as follows. In Section 2, we give some preliminaries including a comparison principle. Then we prove uniqueness of traveling wave with nonzero speed in Section 3.

2. Preliminaries

The following lemma can be easily deduced from (1.5) and (1.6).

Lemma 2.1. Let be a solution of (1.1)–(1.6). If , then as and as for each .

We can determine the sign of the speed (when ) as follows.

Lemma 2.2. Suppose that , then has the same sign as .

Proof. For , an integration by parts gives Then , Therefore, Sending to , using (1.5) and Lemma 2.1, it follows that Hence, the lemma follows.

As a simple consequence of Lemma 2.2, we have if is of balanced type, that is, . Notice that we cannot guarantee the speed is zero or not by using the method developed in [16]. In fact, for the 1D lattice case, the classical work of Keener [21] indicates that the propagation failure (i.e., ) occurs when the diffusion coefficient is sufficiently small, even when is of unbalanced type. A similar result for 1D periodic case can be found in [22]. For our model, the problem for the propagation failure is still open.

Set . Define . Then we have the following comparison principle.

Lemma 2.3. Assume that , and . Suppose that and are bounded and continuous on such that then for all , . Moreover, if there exists some with such that , then for all , .

Since the proof is quite similar to the one given in [16, Lemma  1], we safely omit it here.

3. Uniqueness

In this section, we will study the uniqueness of traveling waves of (1.1)–(1.6). Firstly, applying a method of Fife-McLeod [20], we can derive the following result.

Lemma 3.1. Suppose that is a solution of (1.1) for , such that and for all . If  is a traveling wave solution with speed , then there exist a sufficiently large positive integer and , depending on the initial value, and positive numbers (depending on the value of near and ), , such that for all , , if , , for some small .

Proof. We will only consider the case when . In this case, we have for all .
First, we let
Clearly, is continuous. Fixing with there exists , such that for all . Since is continuous, we would find , such that for all , . By the same reasoning, there exist and , such that for all , .
Define , where
Choose satisfying , and let , be determined later. We claim that for some and . Since , there exists , such that Moreover, since , there exist and , such that Combining (3.8) and (3.9), we have proved the claim (3.7).
Now, we prove . If , then Divide into three cases.
Case 1. . From the above discussion, since , we have Since ,. Choosing , we have in this case.Case 2. . As in Case 1, we have such that if .Case 3. . If and , then . Choosing , we have in this case.Then for all cases. Hence, the second inequality of (3.2) follows from a comparison principle. By the same way, we have the first inequality of (3.2). This proves the lemma.

Note that we have the following different type of super- and sub-solutions which can be verified by a similar way as that of Lemma 3.1.

Lemma 3.2. Suppose that is a solution of (1.1)–(1.6). For any , , , is a small number, and , and let , where then .

We now prove the following uniqueness result.

Theorem 3.3. Suppose that , are two traveling wave solutions of (1.1)–(1.6) with , , then one has and for some .

Proof. As before, we only consider the case when both and are positive. Since by Lemma 3.1, we have for all , with the constants defined in Lemma 3.1.
Let , , and , . Take . We get for all , , By the property (1.5) for all , ,
Setting the moving coordinate we have for any , for all , Suppose that . We may assume that . Fixing and sending , this leads that either or , which is a contradiction. Hence, .
We now suppress the dependence of , and we obtain
For , we have Define Since due to , we have . Assume that . By the strong comparison principle, we know that Since , there exists such that where is the number mentioned in Lemma 3.2. If , by the continuity of , we would find , , such that If , then Combining (3.24) and (3.25), we get Hence, By Lemma 3.2 and the comparison principle, we have Fixing and sending , This contradicts with the definition of . Hence, .

Hence, we obtain the uniqueness (up to translations) of the traveling wave solution with nonzero speed.

Acknowledgments

This work was partially supported by the National Science Council of the Republic of China under the Grant NSC 100-2115-M-005-001. The authors also thank the referee for some valuable comments and suggestions.

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