Abstract and Applied Analysis

Volume 2012 (2012), Article ID 735675, 9 pages

doi:10.1155/2012/735675

Research Article

Monostable-Type Travelling Wave Solutions of the Diffusive FitzHugh-Nagumo-Type System in

Chih-Chiang Huang

Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan

Received 30 March 2012; Accepted 25 May 2012

Academic Editor: Norimichi Hirano

Copyright © 2012 Chih-Chiang Huang. 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.

This paper is concerned with monostable-type travelling wave solutions of the diffusive FitzHugh-Nagumo-type system (FHN) in for the two components and . By solving in terms of , this system can be reduced to a nonlocal single equation for . When the diffusion coefficients in the system are equal, we construct travelling wave solutions for the non-local equation by the method of super- and subsolutions developed by Morita and Ninomiya (2008) Moreover, we propose a condition for , which is similar to the condition Reinecke and Sweers (1999) used to transform (FHN) into a quasimonotone system.

1. Introduction

In the present paper, we are concerned with the diffusive FitzHugh-Nagumo-type system (FHN) in that is,

where , , and . A typical example of is for . Throughout the paper we assume that is a function in and , , and are bounded in for some large constant . In addition, satisfies .

FHN derived from the Hodgkin-Huxley model is a typical model for excitable media. In many fields, such as physics, chemistry, and biology, FHN has become one of the frequently used-reaction diffusion systems to describe interesting phenomena. The solutions of interest here are traveling wave solutions. Let , then travelling wave solutions of (1.1) satisfy

Over the past decades, this system has been extensively studied. For instance, as , under different assumptions, Systems (1.2) and (1.3) admit standing pulses in [1–3], infinitely many periodic solutions in [3], fronts, back waves in [4, 5] and travelling pulses in [5]. For the higher dimension case , symmetric standing waves were established by Reinecke and Sweers [6] and Wei and Winter [7].

As , if the solutions are assumed to be bounded, (1.2) and (1.3) tend to the single equation

Let be a function which has the property that for some , on and on . In addition to the planar waves, (1.4) admits other types of solutions, including travelling curved fronts , conical shapes and pyramidal shapes in [8–11]. Moreover, Hamel and Roquejoffre [12] established travelling wave solutions of (1.4) in which connect one unstable periodic solution at () and one stable constant solution at (). On the other hand, travelling wave solutions of (1.4) in connecting a unstable one-peak solution at () and a stable constant solution () were obtained by Morita and Ninomiya [13].

In this paper, we use the method of super- and subsolutions developed in [13]. Due to technical restriction, we assume . Since (1.3) is linear, can be solved formally in terms of . With expressed in terms of , Systems (1.2) and (1.3) are reduced to the non-local equation

where we denote by . It is readily seen that if is independent of , then by the uniqueness theorem . As , the asymptotic behaviors of travelling wave solutions of (1.5) formally satisfy

where . Our main purpose is to look for monostable-type travelling wave solutions which connect a stable solution of (1.6) as () and a unstable one as (). Without loss of generality, we may assume that is an unstable solution. Throughout this paper, the following hypotheses are assumed.

(H1) There are two solutions of (1.6) satisfying . Moreover, there exist an eigenvalue and its corresponding eigenfunction with and such that

(H2) There exists no other solution of (1.6) with the property .(H3) for some .(H4) For all small , there exists solutions satisfying ,

for some constant . (H5)

where and .

To simplify the proof of the main theorem in this paper, we modify the nonlinear term such that the minimum and maximum of in are the same as those in . For convenience, we still denote for the new modification of . Let and let satisfy . We state the main theorem as follows.

Theorem 1.1. Assume and (H1)–(H5) hold. Then there exists such that for all , Systems (1.2) and (1.3) admit a pair of smooth solutions which satisfie and the boundary conditions , where .

Remark 1.2. In , when the inequality is reversed, that is, , a result similar to Theorem 1.1 can be proved except that the inequalities and in Theorem 1.1 need to be replaced by and , respectively.

Remark 1.3. In fact, can be weakened to the following assumption:

This condition holds if does not decay faster than as . In this case, if we choose , where , then a similar result can be proved.

It is not easy to find an example which satisfies assumptions even for the case since the stability of the radially symmetric solutions obtained in [6, 7] has not yet been studied. However, we believe that for the structure of System (1.2) and (1.3) are similar to that of (1.5). Accordingly, we extend the result of Theorem 2.1 in [13] to the one in Theorem 1.1.

2. Proof of the Main Theorem

To prove the Theorem 1.1, we use the super- and subsolutions constructed in [13]. By considering the following equation, we construct subsolutions of . Let satisfy

For all , the above boundary value problem admits a unique solution (up to a translation) which is strictly increasing in . Subsolutions of are established as follows.

Lemma 2.1. Let . Then there exists such that for all and .

Proof. Let , then . Indeed, it is easy to see that by the maximum principle and . A straightforward calculation gives

Using the maximum principle, we obtain . Therefore by (H1)

where .

Let . By choosing and using (H3), we obtain . According to the mean value theorem, we have if and if . Therefore if , where as and as . The proof is completed.

In what follows we construct supersolutions of .

Lemma 2.2. Let and , where satisfies and . Then .

Proof. Note that and . Indeed, by the uniqueness theorem we have and

It follows from that

where . The last second inequality is due to

We complete the proof of the lemma.

Let

where .

To show the existences of travelling wave solutions of (1.6), we use the following iteration process:

In the following lemma, we assert that the supersolutions of are greater than or equal to the subsolutions of . Moreover, we show that both and are supersolutions of , which is useful in the proof of iteration process.

Lemma 2.3. Assume and let . Then for all there exists depending on such that for all one has

Proof. For the case we take , then

The last inequality holds by (H3). On the other hand,

where . According to , for some positive constant . By choosing , we obtain

which holds due to assumptions and .

For the case , given we choose and use assumption , then

Moreover,

It is readily seen that . By and ,

Setting , the lemma holds.

To generalize the result of Theorem 2.1 in [13], the nonlocal term of (1.5) needs to be better estimated. More precisely, we point wisely control by the local term such that the iterative sequence is comparable with .

Lemma 2.4. Let be nonnegative and solve for some constant . Assume for some . Then .

Proof. Let and . Then because of and the maximum principle. Our main purpose is to claim . By the assumption of and the definition of , we have

The last inequality follows from the hypothesis of and the nonnegativity of . By the maximum principle, .

As becomes large, we claim that the iterative sequence is increasing.

Lemma 2.5. Assume and , then for all and one has and

Proof. We first claim that for all . Indeed, by Lemmas 2.3 and 2.4 (take and ) we obtain

Therefore Lemmas 2.2 and 2.3 yield

where . According to the maximum principle, . It follows form the proof of that . Therefore . Continuing this process, we have for all by induction.

Next obvert that due to Lemma 2.1. By the maximum principle, . Applying Lemma 2.4 to , we have

Therefore

where . Thus . By induction, the sequence of functions is nondecreasing. On the other hand, obvert that . Therefore by (H5), we obtain

Using Lemma 2.4 again, we have

Then by the maximum principle. Inducting in , we obtain .

Proof of Theorem 1.1. By Lemma 2.5, we define . Following the proof of Theorem 2.1 in [13], and , for all we obtain that is a smooth solution of (1.5), and . Let , then by the maximum principle. We complete the proof of the theorem.

Acknowledgments

C.-C. Huang wishes to express his sincere gratitude to Dr. Li-Chang Hung for his careful reading of the paper and valuable suggestions for improvement, and to his supervisor, Professor Chiun-Chuan Chen for his warm encouragement and invaluable advice throughout his Ph.D. program.

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