Research Article | Open Access
Planar Waves of the Buffered Bistable System
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.
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  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 . 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  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)
Recently, Matano et al.  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 . About the stability of planar wave, also see [8–10]. Just recently, Matano and Nara  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  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 ; 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 , 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 .
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 , 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.
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  and Lemma 3.5 in .
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 , and we omit them here. Now we consider the following problem:
Lemma 7 (see ). 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 , 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 .
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.
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 .
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
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.
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 , and we omit it here.
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,