Abstract and Applied Analysis

Volume 2013 (2013), Article ID 936296, 13 pages

http://dx.doi.org/10.1155/2013/936296

Research Article

Planar Waves of the Buffered Bistable System

Xiaohuan Wang¹ and Guangying Lv1,2

¹College of Mathematics and Information Science, Henan University, Kaifeng 475001, China

²Institute of Contemporary Mathematics, Henan University, Kaifeng, Henan 475001, China

Received 12 June 2013; Accepted 9 October 2013

Academic Editor: Svatoslav Staněk

Copyright © 2013 Xiaohuan Wang and Guangying Lv. 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 the large time behavior of disturbed planar fronts in the buffered bistable system in (). We first show that the large time behavior of the disturbed fronts can be approximated by that of the mean curvature flow with a drift term for all large time up to . And then we prove that the planar front is asymptotically stable in under ergodic perturbations, which include quasiperiodic and almost periodic ones as special cases.

1. Introduction

Traveling waves in excitable systems have been the subject of a vast number of mathematical studies for the pastyears. The basic mathematical theory can be used to describe wave propagation in a wide array of biological and chemical systems. Recently, Tsai and Sneyd [1, 2] studied the following buffered system:

where , , , and are positive constants. We call system (1) the full buffering model. When (), Tsai and Sneyd [1] proved the existence, uniqueness, and stability of traveling wave fronts of system (1). The existence, uniqueness, and stability of traveling wave fronts of system (1) with, were obtained by Tsai and Sneyd [2]. About the buffered system (3), see also [3, 4], for more details.

In this paper, we consider the large time behavior of the Cauchy problem to buffering model in:

with initial value

where,. Throughout this paper, we always assume that. In other words, we consider a special case in the present paper and will study the general case in the further paper.

One of the most interesting and natural questions is the behavior of solutionsas, in particular, the question about the stability of traveling wave fronts. Lv [5] studied the system (2) with and obtained the asymptotic stability of planar waves on, where. Under initial perturbation that decays at space infinity, they also proved that the perturbed solution converges to planar waves as . We remark that one can obtain the similar results to Theorems 1 and 3 when . For convenience, we only consider the case that.

It is well known that (2) onhas a traveling wave solution with the form for some constant, connecting two equilibrium and , under the condition thatis of bistable. Here “is bistable” is meant that and , where . A typical example is that , where .

A traveling wave front of (2) is a monotone solution with the form ( is the wave speed), satisfying

where, and .

The functionis also a traveling wave front for system (2) with. We call it planar front. Its translation, whereis any constant, is also called a planar front. Note that the equations in (2) from second to the last have the same properties; without loss of generality, we only consider the case that, and we writeas, that is,

with initial value

Let and , then (5) becomes (for simplicity, we omit the symbol)

Obviously, system (7) is a cooperative model. The problem corresponding to traveling wave front of system (7), connecting and , is

Recently, Matano et al. [6] considered the following Allen-Cahn equation:

Under the condition thatis of the bistable type, they obtained the stability of planar fronts under any-possibly large-initial perturbations and almost periodic perturbations. Whenis monostable type, the stability of planar waves for (9) was obtained by Lv and Wang [7]. About the stability of planar wave, also see [8–10]. Just recently, Matano and Nara [11] reconsidered the Cauchy problem (9) under the condition thatis bistable type. They proved that the planar front is asymptotically stable inunder spatially ergodic perturbation and that the large time behavior of the disturbed planar front can be approximated by that of the mean curvature flow with a drift term for all large time up to. Lv [12] studied the Cauchy problem (9) under thatis monostable type and obtained that the planar front is asymptotically stable inunder spatially ergodic perturbation.

Our work has been inspired by [11]; in this paper, we will consider the stability of planar fronts under spatially ergodic perturbation. Let us now state our main results.

Theorem 1 (large time behavior). Let. Letbe a solution problem (7) with (6) whose initial valuesatisfies

where andare defined as in Lemma 4. Then there exist a constant and a smooth functionsuch that

(i)for each and , one hasif and only if;(ii)it holds that

(iii)for any, there existssuch that the solutionof the problem

satisfies

wheredenotes the-dimensional gradient.

The statement (ii) of Theorem 1 implies that the solution behaves like the function for large; thus the large time behavior of the solution is basically determined by the position of the-level set . The last statement shows that the behavior ofcan be approximated by the solutionof the mean curvature flow onwith a drift term. Comparing the above theorem with Theorem  1.1 in [5], it is to see that we delete the assumptions that initial perturbations decay to zero as, and thus the result in this paper is better than that of [5].

In order to obtain the stability of planar wave, we need the following definition.

Definition 2 (unique ergodicity in the -direction). A bounded uniformly continuous function : is called uniquely ergodic in the-direction if there exists a unique probability measure on , that is, -invariant for any,

andstands for the closure of a setin the-topology.

Theorem 3. In addition to the assumptions of Theorem 1, assume further that and are uniquely ergodic in the-direction. Then there exist constantssuch that

From [11], we see that

wheredenote, respectively, the sets of periodic functions, quasiperiodic functions, almost periodic functions, strictly ergodic functions, and uniquely ergodic functions. Hence, the above Theorem 3 is a general result.

The rest of this paper is organized as follows. In Section 2, some known results and related problem are considered. Section 3 is concerned with the proofs of Theorems 1 and 3.

2. Some Known Results and Related Problem

In this section, we first study the one-dimensional problem (7) in a moving frame, that is,

where.

The traveling wave solutions of (17) have been studied by many authors; see [1, 2], for more details. Tsai and Sneyd [1, 2] obtained the existence, uniqueness, and stability of traveling wave fronts of system (17). They obtained the following lemmas; see Lemma  6.1 in [2] and Lemma  3.5 in [5].

Lemma 4. Assume thatis bistable andis a traveling wave solution of (17). Then there exist , , and such that, for any , the function defined by

is a subsolution. More precisely, it satisfies the following inequality:

Lemma 5. Assume thatis bistable and is a traveling wave solution of (17). Then there exist , , and such that, for any , the function defined by

is a subsolution. More precisely, it satisfies the following inequality:

The proof of the above two lemmas is similar to that of Lemmas 3.5 and 3.6 in [5], and we omit them here. Now we consider the following problem:

Lemma 6 (see [11]). Let. Letbe a solution to the problem (22) whose initial valueis bounded, Lipschitz continuous, and uniquely ergodic on. Then there exists a constantsuch that

Lemma 7 (see [11]). Let and denote the solutions of the following equations:

under the initial conditions. Then, for any constant, there exists a constantsuch that if, it holds that

The following lemma is a little different from Lemma  3.10 in [11], but the proof is similar.

Lemma 8. Letbe a solution to the problem

Then the following estimates hold:

for each , where , , , , and are positive constants such that

(i)depends only on and ;(ii)depends only on,, and and satisfies

(iii)depends only on,, and and satisfies

(iv) and depend only on and .

3. Proofs of Theorems 1 and 3

In this section, we consider problem (7) in and prove our main results. Firstly, we give rough upper and lower bounds for the solution at large time, then introduce the notion of-limit points of the solution and study basic properties of-level surface of the solution. Lastly, we construct a fine set of supersolutions and subsolutions and give the proofs of the main theorems.

We will express solutionsof (1) in a moving frame, and thus the planar waves can be viewed as stationary states. Letting

problem (7) with (6) is rewritten as (we writeasfor simplicity)

where. Throughout this section, we always assume that the initial valuesatisfiesand

whereandare defined in Theorem 1. We first consider upper and lower bounds of the solution of (31) at large time.

Lemma 9. Letbe a solution of (31). Then there exists constant , such that

Proof. We only show the upper bound of, since the other is similar. Letbe as in Lemma 4. Then it suffices to show that there exist constantsandsuch that

Indeed, the comparison principle and (34)-(35) give for, which yield upper bound by letting.

Since, we see from the comparison principle that

Note that, there exists a constantsuch that, for eachand,

Next, we show that

for each. For this purpose, choose positive constants , , and such that

and that

Then the functions

are a supersolution of (31) ifis chosen sufficiently large. Hence,

This proves (38). Combining (37) and (38), it is easy to see that (34)-(35) hold. This completes the proof.

Now, we introduce the notion of-limit points of the solutionof (31), where we consider a sequence both inand. Then we show that any-limit point is a planar wave under the assumption (34).

Definition 10. A vector-valued functiondefined onis called a-limit point of the solutionof (31) if there exists a sequence such thatand that

The following remark tells us how to construct-limit point, which is similar to Remark  4.4 in [11].

Remark 11. Let be a solution of (31). Then for any sequencewith, there exist a subsequence and a -limit point of such that

Indeed, by the assumption in Theorem 1, it is easy to see thatis bounded. Then by-estimates and Schauder’s estimate (see [13, 14]), the solutionbelongs tofor any. Furthermore,

whereis a constant independent of. Letbe a sequence of compact subsets of satisfying

Then, for each, the sequence of functionsis defined onfor all largeand the restrictions of these functions ontoare relatively compact inby virtue of the estimate (45). By using diagonal argument, we can choose a subsequenceand a functiondefined onsuch that, for any, it holds that

which meansasin.

In order to prove that any-limit point is a planar wave, the following Lemma is needed.

Lemma 12. Letbe a vector-valued function that is defined onand satisfies

Assume further that there exist two constants such that

Then there exists a constantsuch that

Proof. Define

Then,

It follows from the monotonicity ofandthat there existssatisfying

Indeed, let, then. Now, let

Since, and by using (49), there exists constantsuch that

Note that one can assume that. Let, then for and for . It is easy to see that and . Assume that and call . If and , then there exists such that in for all . Denote , and , we see that on and satisfies

for some constant(remember that andandare bounded). The parabolic maximum principle implies thatin. Similarly, we can prove thatfor. Therefore,infor all. This contradicts the minimality of. It follows then that

We first consider the case thatand. As a consequence, there existand a sequencesuch that

Calland. Note thatis a bounded sequence. Up to extraction of a subsequence, the functionsconverge locally uniformly to a solutionof the following system:

It is easy to see that

and ; that is, attains the local minimum at . We only consider the equation of . It is easy to verify thatcannot attain the local minimum because

So we get a contradiction. Similarly, one can deal with the other case: and .

Now we consider the last case:and. As the case thatand, we obtain a solutionof system (59) and. It follows from the strong parabolic maximum principle thatfor alland theninby uniqueness of the solution of the Cauchy problem (31). Thus, for all. Butassince. This is a contradiction.

Thus,,

Sinceandare arbitrary, we conclude thatanddepend ononly, namely,for some. Note thatis a solution of (59), we conclude that. This completes the proof.

From Lemma 9, any-limit point ofsatisfies

for some constant . Combining the above lemma, we immediately have the following result.

Lemma 13. Let be a solution of (31). Then any-limit point of is a planar wave; that is, there exists a constant such that

Now, we derive estimate for the derivatives of the solution of (31).

Lemma 14 (monotonicity in). Letbe a solution of (31). Then for any constant , there exists a constantsuch that

Proof. We prove this lemma by contradiction. If the above claim does not hold, then there exists a sequencesuch that, and that

Replaceby its subsequence if necessary; we may assume without loss of generality thatconverges to some limitand that

whereis a limit point of. Hence,

On the other hand, Lemma 13 shows thatfor some. This gets a contradiction because. The proof of the lemma is completed.

By using Lemmas 14 and 9, the following corollary is obtained.

Corollary 15. Letbe a solution of (31). Then there exists a constantsuch that

where

Lemma 16 (decay of-derivatives). Letbe a solution of (31). Then for any constant, it holds that

for each.

The proof of this lemma is similar to Lemma  4.9 in [11], and we omit it here.

Next we study the-level surface of the solution of (31). From Corollary 15 and Lemma 16, we can derive the following lemma that the-level surface of the solutionhas a graphical representationfor all.

Lemma 17. Letbe a solution of (31) and letbe as defined in Corollary 15. Then there exists a smooth bounded functionsuch that

for any. Furthermore, the following estimates hold:

(i)for each,

(ii)there exists a constantsuch that, for each,

Proof. Since

is bounded in the-direction by virtue of Lemma 9 and the factsand, we can define a bounded functionsatisfying (72) thanks to Corollary 15. Hereis smooth by the implicit function theorem, sinceis smooth for. The other estimates follow from Lemma 16, and we omit it here. This completes the proof.

The following lemma shows that the large time behavior of the solution can be essentially determined by the-level surface.

Lemma 18. Letbe a solution of (31) and letbe as defined in Lemma 17. Then, it holds that

Proof. We prove this lemma by contradiction, and we only consider. If the above claim does not hold, there exist a constantand a sequencesuch that and that

On the other hand, by virtue of Lemma 9 and boundedness of, we can choose constants and such that

which means thatis bounded. We can choose subsequence of, which we denote again bysuch that

whereis some-limit point of. This and (77) show that

On the other hand, we have

Lemma 13 implies that. This contradicts (80). This completes the proof of this lemma.

By settingand, we obtain the statement (i), (ii) of Theorem 1 from Lemmas 17 and 18. Thus, it remains to prove the statement (iii). This will be done at the end of this section.

In the following, we construct supersolutions of (31). For this purpose, we consider the problem of the form

Lemma 19 (supersolution). For any constant , there exist positive constant , , and smooth functionsandsatisfying

where, such that ifis any solution of (82) withand, then the function defined by

satisfies

Proof. We divide the proof into four steps. The first two steps are similar to that of Lemma  4.12 in [11], and we only give the last results.

Step 1. Set

Then, by using (8), we have

By rewriting the above expression in terms of, we obtain

where, (; ) are functions given by

Step 2. Now we estimate (). Define and by

It follows from Theorem  2.1 in [5] that , , and decay to zero exponentially as . Noting that is assumed to be bounded, the following functions are all bounded:

Now we choose a constantarbitrarily. Then from the above boundedness and Lemma 8, we can choose a constantdepending only onand, and a constantdepending only onsuch that if, it holds that

Step 3. Now we determine the constant,and the smooth functionsand (). Set

Then, direct calculation shows that

Hence, there exist constants,andsuch that

whereand

It follows that there exist positive constantsandsuch that for all, we have

This implies that

Denote. Using the above estimates and the facts that, , and, there exist constantssuch that

Set

We define the constantandby

We choose functionssatisfying

Then (107) holds, since we have

Step 4. Now we complete the proof. Sinceandby Step 2, it suffices to show the inequality (). Letting , when, we have

since we have

For any, we have

In summary, we haveand. This completes the proof.

Lemma 20 (subsolution). For any constants, there exist positive constant , , and smooth functions and satisfying

where, such that ifis any solution of (82) withand, then the function defined by

satisfies

The proof of Lemma 20 is similar to that of Lemma 19, and we omit it here.

Lemma 21 (approximation of). Letbe a solution of (31) and letbe as defined in Lemma 17. Then for any, there exists a constantsuch that the functiondefined by

satisfies

Proof. From Corollary 15, we can choose constants,, andsuch that

For the constantand , we choose a constantand functionssatisfying

as in Lemma 19. Then it follows from Lemma 17 that there exists a constant such that . Moreover, using Lemma 18, by choosing larger if necessary, we have

where we used the smallness of. For such, we defineas a function satisfying (110). Then Lemma 19 shows that the function defined by

is a supersolution of (31) for. Since (114) implies that and , the comparison principle gives that and for . Then, due to , we have

Noting that, we get

which implies thatfor. Similarly, by using the subsolutiongiven in Lemma 20, we can show thatfor. This completes the proof.

Now we are ready to complete the proof of Theorem 1.

Proof of Theorem 1. The statements (i) and (ii) of Theorem 1 are directly from Lemmas 17 and 18, respectively. Thus, we only prove the statement (iii).

By Lemma 21, the large time behavior of the-level surfaceof the solutionof (31) can be approximated by functionof the equation

This means that the-level surfaceof the solutionof (7) can be approximated by functionof the equation

Hence, the statement (iii) of Theorem 1 follows from Lemma 21. This completes the proof of Theorem 1.

Next, we prove Theorem 3. Firstly, similar to Lemma  4.15 in [11], one can prove that the-level surfaceremains uniquely ergodic for all large.

Lemma 22 (ergodicity of-level surface). Letbe a solution of (31) and assume thatis uniquely ergodic in the-direction. Then the-level surfacedefined in Lemma 17 is uniquely ergodic for each, whereis the constant in Lemma 17.

Proof of Theorem 3. Letandbe as in Lemma 17, thenis uniquely ergodic for eachfrom Lemma 22. Combining Lemmas 6 and 7, one can easily prove Theorem 3.

Acknowledgments

The first author was supported in part by PRC Grants NSFC 11226168 and 11301146, and the second author was supported in part by PRC Grants NSFC 11171064, 11301146, and 11226168 and the Natural Science Foundation of Jiangsu province BK2011583. The authors are grateful to the referees for their valuable suggestions and comments on the original paper.

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