Journal of Mathematics
Volume 2017 (2017), Article ID 5328246, 9 pages
https://doi.org/10.1155/2017/5328246

Research Article

Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in

Nai-Wei Liu1,2

¹School of Mathematics, Shandong University, Jinan, Shandong 250100, China
²School of Mathematics and Information Science, Yantai University, Yantai, Shandong 264005, China

Correspondence should be addressed to Nai-Wei Liu; liunaiwei@aliyun.com

Received 10 February 2017; Accepted 4 April 2017; Published 7 May 2017

Academic Editor: Emir Köksal

Copyright © 2017 Nai-Wei Liu. 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 consider the interaction of traveling curved fronts in bistable reaction-diffusion equations in two-dimensional spaces. We first characterize the growth of the traveling curved fronts at infinity; then by constructing appropriate subsolutions and supersolutions, we prove that the solution of the Cauchy problem converges to a pair of diverging traveling curved fronts in under appropriate initial conditions.

1. Introduction

In the current paper, we consider the following Cauchy problem:

where is a bounded initial function and the function is of bistable type. Concretely, we assume that satisfies the following:

(F1) and , and ,(F2) and for ; and for ,(F3).

Such an example is

Under assumptions (F1) and (F2), it is clear that there exists a positive constant such that

where .

We remark here that the steady states and of (1) are asymptotically stable if (F1)–(F3) hold. In , by letting and , one has

It is well known that (4) has a planar traveling wave front which is unique up to phase shift under assumptions (F1)–(F3), with the unique positive traveling wave speed . The traveling wave fronts have been widely studied in many fields, such as biology, chemistry, epidemiology, and physics. One can refer to [1–8] for details. Traveling wave fronts are special solutions of (1), which can be used to characterize the invariant set with respect to transition in spaces.

Without loss of generality, assume that the solution of (1) travels towards -direction and let

then, we can rewrite (1) into

For convenience, we denote the solution of Cauchy problem (6) with initial function by .

Considering the traveling wave fronts of (1) with traveling wave speed in -direction, then

It is obvious that the solution of (7) is a stationary solution of (6). Let ; then is a solution of (7), where is a solution of (4) and

We call the solution of (7) traveling curved front, since it is nonplanar. By using comparison principle, one has the function

which is a subsolution of (7) with on . By using sub- and supersolutions method, Ninomiya and Taniguchi [9, 10] proved the existence and global stability of traveling curved fronts for (1).

Theorem 1 (see [9, 10]). Assume that (F1)–(F3) hold. For any , there exists a traveling curved front of (1) such that

Furthermore, if , there is a constant such that

If satisfies

then

where is the solution of the Cauchy problem (1).

It follows from Theorem 1 that (1) has a unique traveling curved front for each , which is globally stable in the sense of (14). In fact, there are many mathematical models arising in biology, population dynamics, flame propagation, and disease spread which can be described by traveling curved front. For example, Sheng et al. [11] considered the stability of traveling curved fronts (V-shaped) for Allen-Cahn equations, and they in [11] also proved that the traveling curved fronts (V-shaped) are not asymptotically stable under some perturbations. In another paper, by using comparison principle, Sheng [12] studied the existence and stability of time-periodic traveling curved fronts about bistable reaction-diffusion equations in . In [13], Wang and Bu considered traveling curved fronts (nonplanar) for combustion and degenerate Fisher-KPP type reaction-diffusion equations. Ninomiya and Taniguchi [9, 10] and Taniguchi [14, 15] showed the existence and the stability of traveling curved fronts for Allen-Cahn equations. Furthermore, by constructing some appropriate subsolutions and supersolutions, Hamel et al. [16] considered the existence and the global stability of traveling curved fronts for a model about conical flames. They in [17] established the existence of traveling curved fronts for bistable model by introducing the conical-limiting conditions at infinity. For more interesting results about the existence and stability of traveling curved fronts, one can refer to [18–28].

In addition to the stability results about traveling fronts mentioned above, the interaction between traveling fronts is also an important topic for reaction-diffusion equations. Here, the interaction of traveling fronts means that the solutions of the Cauchy problem converge to a pair of diverging traveling fronts. Recently, there are many results about this problem. Particularly, Fife and McLeod [29, 30] studied the interaction of traveling fronts in one-dimensional space when . Indeed, they in [29, 30] proved that the solutions of the Cauchy problem converge to a single traveling front, a pair of diverging traveling fronts, and a stacked combination of traveling fronts in , respectively. Based on comparison principle, Chen [3] developed the squeeze technique to study the interaction and the exponential stability of traveling wave solution for bistable reaction-diffusion equations. Furthermore, Roquejoffre [31] expanded the results in [29] to infinite cylinders. In another paper, Bebernes et al. [32] proved that the solution converges to a pair of diverging traveling fronts in cylindrical domains. We also remark here that there is another form of interaction between traveling fronts, which can be described by the so-called entire solutions. Entire solutions can be used to imply the dynamics of two traveling fronts as ; one can refer to [33–37] for related works.

However, the interaction of traveling curved fronts of reaction-diffusion equations in whole spaces is still open. Since two traveling curved fronts traveling towards opposite directions always interact with each other, a natural issue is that whether we can expect that the solution of (1) converges to a pair of diverging traveling curved fronts in under some appropriate initial conditions, which behaves as the interaction of traveling curved fronts. The current paper is devoted to resolving this problem for bistable reaction-diffusion equations in .

In this paper, based on comparison principle, we first construct appropriate sub- and supersolutions and then show that the solution of (1) converges to a pair of diverging traveling curved fronts, which will be done in Section 3. Before doing those, by using the asymptotic decay of planar traveling wave fronts, we give some asymptotic estimates for traveling curved fronts at infinity and list the main result in Section 2.

2. Preliminaries and Main Result

In this section, we first study the asymptotic behavior of traveling curved front of (1) as by using the result of the exponential convergence of one-dimension traveling wave solution of (4) at infinity. In fact, it follows from [38] that there exist positive constants and such that

where and . From [34], we see that the planar traveling wave front of (4) satisfies

for some and defined above.

Under conditions (F1)–(F3) and (3), there exists a constant with , such that

for with as in (3). Furthermore, by virtue of (12), we have

for . Since the traveling wave front of (4) possesses invariance up to translation, we assume that traveling wave front satisfies

and the constant in (16) satisfies

We take three positive constants , , and satisfying

where

By a translation in the -direction, we next take

where is defined in (21). Then, we have

by view of (16) and (19).

In the following, we consider planar traveling wave front satisfying (24) and (25) instead of the solution of (4) and assume that Theorem 1 holds with instead of in the definition of . For convenience, in the rest of the paper we drop the tilde of and denote also by .

By using the asymptotic behavior of planar traveling wave fronts of (4), we immediately obtain the following lemma.

Lemma 2. Assume that satisfies (F1)–(F3). Then there exist some positive constants , , and , such that the traveling curved front defined in Theorem 1 satisfies

where is defined in (21) and

Furthermore, there is

Proof. It follows from (9), (11), and (23) that

Thus (26) holds for by (24).
Inequality (27) follows from the standard elliptic estimates. Next, we prove that (29) holds. In fact, if (29) is not true, there exist and satisfying

Define

where

with a given constant. By extracting subsequence of and denoting the subsequence also by , we have

where is a solution of (7). On the other hand, by view of (10) and (11), we have

Thus the strong maximum principle implies

Then,

which contradicts the assumption . Thus, we complete the proof.

Our main result is the following.

Theorem 3. For every , let be the traveling curved front of (1) defined in Theorem 1 with speed . Assume that (F1)–(F3) hold. Then if satisfies

there exist positive constants and , such that, for all , the solution of (1) satisfies

Furthermore, one has

locally uniformly with respect to .

Remark 4. Inequality (39) implies that the -profile of approaches that of the traveling curved fronts. In particular, it shows that the domain in which is close to 1 is expanding at the speed of . The phase shift is a positive constant which will be defined in the proof of Theorem 3. The similar stability about traveling curved front in cylinder domain is treated in [32].

In the last of this section, we give the definitions of subsolution and supersolutions for (1) in .

Definition 5. If a function and satisfies

then is called a subsolution for (1) in . Similarly, by reversing the inequality in (41), we can define a supersolution for (1).

3. Proof of Theorem 3

In this section, we prove the main result by constructing appropriate sub- and supersolutions. In the following lemma, we construct a subsolution for (1).

Lemma 6. Assume that (F1)–(F3) hold. Let . Then the function

is a subsolution of (1) on , where is traveling curved front of (1) defined in Theorem 1 and are constants defined in (21).

Proof. Define

By using the above prepared results, direct calculations give

If , we consider two cases and , respectively.
Case A ( ). By virtue of (3), (21), and (25), we have

since . By using (17) and (25) and the fact that for , we have

Thus, we have

In the last inequality, we have used the facts (20), (21), and (26).
By a similar argument, we have for with .
Case B ( ). In a similar way as above, for , we have

where . Particularly, we have (46) in this case. Lastly, due to (18), we have

Combining (46), (48), and (49), we have

Consequently, (21) implies for this case.
Similarly, we can prove when . Thus, we have showed that is a subsolution of (1) on .

In order to construct a supersolution for (1), we introduce the following lemmas.

Lemma 7. Assume that (F1)–(F3) hold. Let ; then for any given constant , the function

is a supersolution of (1) on , where is traveling curved front of (1) as in Theorem 1 and are constants defined in (21).

Proof. As the proof of Lemma 6, we need only to prove that the right hand of (43) is nonnegative for the function for . By a similar argument, direct calculations give

To complete the proof, we consider two cases and , respectively.
For , we just consider the case . Since , it follows from (3) that

By virtue of (11) and (21), we have

For , we have

where . Particularly, (18) implies

Therefore, by (21) we get

Thus, defined by (51) is a supersolution of (1).

In a similar way, we prove the following lemma.

Lemma 8. Assume that (F1)–(F3) hold. Let ; then for any given constant , the function

is a supersolution of (1) on , where is traveling curved front of (1) as in Theorem 1 and are constants defined in (21).

Remark 9. Let and ; it follows from Lemmas 7 and 8 that the function

is a supersolution of (1) on .

To complete the proof of Theorem 3, we establish the following comparison result.

Lemma 10. Let and be defined by (42) and (59), respectively, and satisfies (38). Then, there exists such that

for all .

Proof. By (38) and the definition of (42), when , direct calculations give

for . Similarly, we have for . Therefore, the maximum principle for parabolic equations shows for .
Similarly, by the definition of (59), when

By (38), we have that, for and , there exists such that

in , since .
In the range of , we have . Thus, by choosing and using (20), (21), and (26), we have

for . Consequently, we have in this range.
Combining (63) and (64) and taking , we conclude that for . Then the maximum principle for parabolic equations derives that (60) holds.

Proof of Theorem 3. It follows from Lemma 10 that (60) holds with . Thus, by (11), we obtain

for all , if . On the other hand, for , we have

if and . By a similar argument, for , (66) also holds.
Thus, if satisfies (38), by taking and , we obtain that (39) holds, where and are defined in (21).
The asymptotic behavior (40) immediately follows from (39). This completes the proof of Theorem 3.

4. Discussion

In the current paper, we have proved that the solutions of the bistable reaction-diffusion equations converge to a pair of diverging traveling curved fronts in . It means that the solution of (1) with initial function satisfied (38) behaving as two traveling curved fronts traveling towards opposite directions and approaching each other. Our result is different from the stability results in [9–11, 22, 26, 27]. Indeed, the interaction between traveling wave fronts plays an important role in the study of reaction-diffusion equations in , which is crucially related to the pattern formation problem, and there are important applications in chemical, physical, biological systems; see, for example, [39–41].

At last, we note here that the global exponential stability of traveling curved fronts in the sense of Theorem 3 is a difficult problem, since the level set of the traveling curved fronts of (1) have two asymptotic directions as , and both directions make an angle with the negative -axis, which is different from the case of planar traveling fronts (see [20]). We will leave it for a further study. Moreover, how the solution of (1) approaches a “stacked” combination of traveling curved fronts just as the study in Fife and McLeod [29] is also an interesting problem but remains open.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

The author is very grateful to Dr. Wei-Jie Sheng for helpful discussions. The author’s work was partially supported by NSF of China (11371179 and 11401513) and by China Postdoctoral Science Foundation Funded Project (2014M560546).

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