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Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in [R.sup.2].

1. Introduction

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

[partial derivative]u/[partial derivative]t = [DELTA]u + f (u), (x, y) [member of] [R.sup.2], t > 0, u(x, y, 0) = [u.sub.0](x, y), (x, y) [member of] [R.sup.2],

where [u.sub.0] [member of] [C.sup.1]:([R.sup.2]) is a bounded initial function and the function f is of bistable type. Concretely, we assume that f satisfies the following:

(F1) f [member of] [C.sup.1]:(R) and f(0) = f(1) = 0, f'(0) < 0 and f'(1) < 0,

(F2) f(s) < 0 and f'(s) < 0 for s > 1; f(s) > 0 and f'(s) < 0 for s < 0,

(F3) [[integral].sup.1.sub.0] f(s)ds > 0.

Such an example is

f (u) = u(1 - u) (u - a), for 0< a < 1/2. (2)

Under assumptions (F1) and (F2), it is clear that there exists a positive constant [delta] (0 < [delta] < 1/2) such that

f' (u) [less than or equal to] -[[gamma].sub.1], u [member of] (-[infinity], 2[delta]) [union] (1 - 2[delta], [infinity]), (3)

where [[gamma].sub.1] := (1/2) min{-f'(0), -f'(1)} > 0.

We remark here that the steady states 0 and 1 of (1) are asymptotically stable if (F1)-(F3) hold. In [R.sup.1], by letting u(x-[bar.c]t) = [phi]([xi]) and [xi] = x - [bar.c]t ([bar.c] > 0), one has

[mathematical expression not reproducible]. (4)

It is well known that (4) has a planar traveling wave front [phi]([xi]) which is unique up to phase shift under assumptions (F1)-(F3), with the unique positive traveling wave speed [bar.c]. 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 v-direction and let

u(x, y, t) = V(z, y, t), z = x - ct; (5)

then, we can rewrite (1) into

[V.sub.t] - [V.sub.zz] - [V.sub.yy] - c[V.sub.z] - f(V) = 0, (z, y) [member of] [R.sup.2], t > 0, v[|.sub.t=o] = [u.sub.0], (z, y) [member of] [R.sup.2].

For convenience, we denote the solution of Cauchy problem (6) with initial function V(z, y, 0; [u.sub.0]) = [u.sub.0](z, y) by V(z, y, t; [u.sub.0]).

Considering the traveling wave fronts u(x, y, t) = [PHI](z, y) (z = x - ct) of (1) with traveling wave speed c in x-direction, then

-[[PHI].sub.zz] - [[PHI].sub.yy] - c[[PHI].sub.z] - f([PHI]) = 0, in [R.sup.2]. (7)

It is obvious that the solution of (7) is a stationary solution of (6). Let c [greater than or equal to] [bar.c]; then [phi](([bar.c]/c)(z [+ or -] [m.sub.*]y)) is a solution of (7), where [phi](*) is a solution of (4) and

[m.sub.*] := [square root of ([c.sup.2] - [[bar.c].sup.2])]/[bar.c] > 0. (8)

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

[phi](z,y) := max{[phi]([bar.c]/c(z - [m.sub.*] y)), [phi]([bar.c]/c(z + [m.sub.*] y))} = [phi]([bar.c]/c(z - [m.sub.*] [absolute value of (y)])) (9)

which is a subsolution of (7) with [[phi].sub.z](z, y) < 0 on [R.sup.2]. 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 c > [bar.c], there exists a traveling curved front u(x, y, t) = [PHI](z, y) (z = x - ct) of(1) such that

[mathematical expression not reproducible], (10)

[mathematical expression not reproducible]. (11)

Furthermore, if 0 < [[delta].sub.1] < 1/2, there is a constant [gamma]([[delta].sub.1]) > 0 such that

[[PHI].sub.z] (z, y) [less than or equal to] -[gamma]([[delta].sub.1]), for [[delta].sub.1] [less than or equal to] [PHI] [less than or equal to] 1 - [[delta].sub.1]. (12)

If [u.sub.0](z, y) satisfies

[mathematical expression not reproducible], (13)

then

[mathematical expression not reproducible], (14)

where u(x, y, t; [u.sub.0]) is the solution of the Cauchy problem (1).

It follows from Theorem 1 that (1) has a unique traveling curved front [PHI](x - ct, y) for each c > [bar.c], 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 [R.sup.3]. 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 t [right arrow] +[infinity]. 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 [R.sup.1], 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 t [right arrow] -[infinity]; one can refer to [33-37] for related works.

However, the interaction of traveling curved fronts of reaction-diffusion equations in whole spaces [R.sup.2] 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 [R.sup.2] 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 [R.sup.2].

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 [PHI](z, y) of (1) as z [right arrow] -[infinity] by using the result of the exponential convergence of one-dimension traveling wave solution [phi]([xi]) of (4) at infinity. In fact, it follows from [38] that there exist positive constants A and B such that

[mathematical expression not reproducible], (15)

where [mathematical expression not reproducible]. From [34], we see that the planar traveling wave front [phi] of (4) satisfies

1 - [phi] ([xi]) [less than or equal to] k[e.sup.[??][xi]], [xi] [less than or equal to] 0 (16)

for some k > 0 and [??] defined above.

Under conditions (F1)-(F3) and (3), there exists a constant [[gamma].sub.2] with [[gamma].sub.2] > [[gamma].sub.1] > 0, such that

f(u) [less than or equal to] [[gamma].sub.2] (1 - u) (17)

for 0 < 1 - u [less than or equal to] [delta] with [delta] as in (3). Furthermore, by virtue of (12), we have

[[PHI].sub.z] (z, y) [less than or equal to] -[gamma] ([delta]) := -[[gamma].sub.3] < 0 (18)

for [delta] [less than or equal to] [PHI] [less than or equal to] 1 - [delta]. Since the traveling wave front [phi]([xi]) of (4) possesses invariance up to translation, we assume that traveling wave front [phi]([xi]) satisfies

[phi](0) = [theta], [theta] [greater than or equal to] 1 - [delta]/2 (19)

and the constant k in (16) satisfies

k [less than or equal to] min{[[gamma].sub.1][delta]/4([[gamma].sub.2] - [[gamma].sub.1]), [delta]/4}. (20)

We take three positive constants [q.sub.0], [mu], and M satisfying

[delta]/4 [less than or equal to] [q.sub.0] [less than or equal to] [delta]/2, 0 < [mu] [less than or equal to] min{[[gamma].sub.1]/2, [lambda]c}, m [greater than or equal to] max{[M.sub.1], [M.sub.2]}, (21)

where

[mathematical expression not reproducible]. (22)

By a translation in the [xi]-direction, we next take

[??]([xi]) = [phi]([xi] - M), (23)

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

[mathematical expression not reproducible], (24)

[??](M) = [phi](0) = [theta], [theta] [greater than or equal to] 1 - [delta]/2, (25)

by view of (16) and (19).

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

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

Lemma 2. Assume that f satisfies (F1)-(F3). Then there exist some positive constants [lambda], k, and C, such that the traveling curved front [PHI](z, y) defined in Theorem 1 satisfies

[mathematical expression not reproducible], (26)

[absolute value of ([[PHI].sub.Z] (z, y))] [less than or equal to] C, (z, y) [member of] [R.sup.2], (27)

where M is defined in (21) and

[lambda] := [bar.c]/c [??] = [bar.c][square root of ([[bar.c].sup.2] - 4f' (1) - [[bar.c].sup.2])]/2c > 0. (28)

Furthermore, there is

[mathematical expression not reproducible]. (29)

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

[mathematical expression not reproducible]. (30)

Thus (26) holds for z [less than or equal to] 0 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 [[epsilon].sub.1] > 0 and [{([z.sub.n], [y.sub.n])}.sup.[infinity].sub.n=1] satisfying

[mathematical expression not reproducible]. (31)

Define

[[PHI].sub.n] (z, y) = [PHI] (z + [z.sub.n], y + [y.sub.n]), in [B.sub.0], (32)

where

[B.sub.0] := {(z, y) [member of] [R.sup.2] | [z.sup.2] + [y.sup.2] < C}, (33)

with C > 0 a given constant. By extracting subsequence of [[PHI].sub.n] and denoting the subsequence also by [[PHI].sub.n], we have

[[PHI].sub.n] (z, y) [right arrow] [PHI]* (z, y) in [C.sup.2.sub.loc] ([R.sup.2]), (34)

where [PHI]*(z, y) is a solution of (7). On the other hand, by view of (10) and (11), we have

[mathematical expression not reproducible]. (35)

Thus the strong maximum principle implies

[partial derivative]/[partial derivative]z [PHI]* (z, y) = 0, in [B.sub.0]. (36)

Then,

[partial derivative]/[partial derivative]z [[PHI].sub.n] (z, y) [right arrow] 0, as n [right arrow] [infinity], (37)

which contradicts the assumption [[PHI].sub.z]([z.sub.n], [y.sub.n]) [greater than or equal to] [[epsilon].sub.1] > 0.

Thus, we complete the proof.

Our main result is the following.

Theorem 3. For every c > [bar.c], let [PHI](*, y) be the traveling curved front of (1) defined in Theorem 1 with speed c. Assume that (F1)-(F3) hold. Then if [u.sub.0](x, y) [member of] (0, 1) satisfies

[mathematical expression not reproducible], (38)

there exist positive constants q, [mu] > 0 and [zeta], such that, for all (x, y, t) [member of] [R.sup.2] x [0, +[infinity]), the solution u(x, y, t; [u.sub.0]) of (1) satisfies

[mathematical expression not reproducible]. (39)

Furthermore, one has

[mathematical expression not reproducible] (40)

locally uniformly with respect to (x, y) [member of] [R.sup.2].

Remark 4. Inequality (39) implies that the x-profile of u(x, y, t; [u.sub.0]) approaches that of the traveling curved fronts. In particular, it shows that the domain in which u is close to 1 is expanding at the speed of c. The phase shift [zeta] 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 [R.sup.2] x (0, +[infinity]).

Definition 5. If a function [u.bar](x, y, t) [member of] [C.sup.2,1]([R.sup.2] x (0, +[infinity]), R) and satisfies

[partial derivative][u.bar](x, y, t)/[partial derivative]t [less than or equal to] [DELTA][u.bar](x, y, t) + f([u.bar](x, y, t)), (x, y) [member of] [R.sup.2], t > 0, (41)

then [u.bar](x, y, t) is called a subsolution for (1) in [R.sup.2] x (0, +[infinity]). Similarly, by reversing the inequality in (41), we can define a supersolution [bar.u](x, y, t) 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 c > [bar.c]. Then the function

[[phi].bar.](x, y, t) := [PHI](x - ct + M (1 - [e.sup.-[mu]t]), y) + [PHI](-x - ct + M (1 - [e.sup.-[mu]t], y) - 1 - [q.sub.0][e.sup.-[mu]t] (42)

is a subsolution of (1) on t [member of] (0, [infinity]), where [PHI](*, y) is traveling curved front of (1) defined in Theorem 1 and [q.sub.0], [mu] > 0 are constants defined in (21).

Proof. Define

F([[phi].bar](x, y, t)) := [[phi].bar](x, y, t)/[partial derivative]t - [DELTA][[phi].bar](x, y, t) - f([[phi].bar](x, y, t)). (43)

By using the above prepared results, direct calculations give

[mathematical expression not reproducible]. (44)

If x [greater than or equal to] 0, we consider two cases [PHI](x - ct + M (1 - [e.sup.-[mu]t]), y) [member of] [0, [delta]] [union] [1 - [delta], 1] and [PHI](x - ct + M (1 - [e.sup.-[mu]t]), y) [member of] [[delta], 1 - [delta]], respectively.

Case A ([PHI](x - ct + M (1 - [e.sup.-[mu]t]), y) [member of] [1 - [delta], 1]). By virtue of (3), (21), and (25), we have

[mathematical expression not reproducible], (45)

since 0 < 1 - [PHI](-x - ct + M(1 - [e.sup.-[mu]t]), y) + [q.sub.0][e.sup.-[mu]t] [less than or equal to] 1 - [phi](M) + [q.sub.0] [less than or equal to] [delta]. By using (17) and (25) and the fact that 0 < 1 - [PHI](-x - ct + M(1 - [e.sup.-[mu]t]), y) [less than or equal to] 1 - [phi](M) [less than or equal to] [delta]/2 for x [greater than or equal to] 0, we have

f([PHI](-x - ct + M(1 - [e.sup.-[mu]t]), y)) [less than or equal to] [[gamma].sub.2] (1 - [PHI](-x - ct + M(1 - [e.sup.-[mu]t]), y)). (46)

Thus, we have

[mathematical expression not reproducible]. (47)

In the last inequality, we have used the facts (20), (21), and (26).

By a similar argument, we have F([[phi].bar]) [less than or equal to] 0 for [PHI](x - ct + M(1 - [e.sup.-[mu]t]), y) [member of] [0, [delta]] with x [greater than or equal to] 0.

Case B ([PHI](x - ct + M(1 - [e.sup.-[mu]t]), y) [member of] [[delta], 1 - [delta]]). In a similar way as above, for x [greater than or equal to] 0, we have

[mathematical expression not reproducible], (48)

where [L.sub.2] := [max.sub.-[delta] [less than or equal to] u [less than or equal to] 1-[delta]][absolute value of (f'(u))]. Particularly, we have (46) in this case. Lastly, due to (18), we have

[[PHI]'.sub.z](x - ct + M(1 - [e.sup.-[mu]t]), y) + [[PHI]'.sub.z](-x - ct + M(1 - [e.sup.-[mu]t]), y) [less than or equal to] -[[gamma].sub.3]. (49)

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

[mathematical expression not reproducible]. (50)

Consequently, (21) implies F([[phi].bar]) [less than or equal to] 0 for this case.

Similarly, we can prove F([[phi].bar]) [less than or equal to] 0 when x < 0. Thus, we have showed that [[phi].bar](x, y, t) is a subsolution of (1) on t [member of] [0, [infinity]).

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

Lemma 7. Assume that (F1)-(F3) hold. Let c > [bar.c]; then for any given constant N > 0, the function

[[phi].sup.1.sub.N](x, y, t) := [PHI](x - ct - M (1- [e.sup.-[mu]t]) - N, y) + [q.sub.0][e.sup.-[mu]t] (51)

is a supersolution of (1) on t [member of] (0, +[infinity]), where [PHI](*, y) is traveling curved front of (1) as in Theorem 1 and [q.sub.0], [mu] > 0 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 [[phi].sup.1.sub.N](x, y, t) for (x, y, t) [member of] [R.sup.2] x (0, +[infinity]). By a similar argument, direct calculations give

[mathematical expression not reproducible]. (52)

To complete the proof, we consider two cases [PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) [member of] [0, [delta]] [union] [1 - [delta], 1] and [PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) [member of] [[delta], 1 - [delta]], respectively.

For [PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) [member of] [0, [delta]] [union] [1 - [delta], 1],we just consider the case [PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) [member of] [1 - [delta], 1].

Since [PHI](x - ct - M(1- [e.sup.-[mu]t]) - N, y) + [q.sub.0][e.sup.-[mu]t] [member of] [1 - [delta], 1 + [delta]], it follows from (3) that

f([PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) + [q.sub.0][e.sup.-[mu]t]) - f([PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y)) [less than or equal to] -[[gamma].sub.1][q.sub.0] [e.sup.-[mu]t]. (53)

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

[mathematical expression not reproducible]. (54)

For [PHI](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) [member of] [[delta], 1 - [delta]],we have

[mathematical expression not reproducible], (55)

where [L.sub.3] = ma[x.sub.-][delta] [less than or equal to] u [less than or equal to] 1+[delta] [absolute value of (f'(u))]. Particularly, (18) implies

[[PHI]'.sub.z](x - ct - M(1 - [e.sup.-[mu]t]) - N, y) [less than or equal to] -[[gamma].sub.3]. (56)

Therefore, by (21) we get

[mathematical expression not reproducible]. (57)

Thus, [[phi].sup.1.sub.N](x, y, t) 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 c > [bar.c]; then for any given constant N > 0, the function

[[phi].sup.2.sub.N](x, y, t) := [PHI](-x - ct - M(1 - [e.sup.-[mu]t]) - N, y) + [q.sub.0][e.sup.-[mu]t] (58)

is a supersolution of (1) on t [member of] (0, +[infinity]), where [PHI](*, y) is traveling curved front of (1) as in Theorem 1 and [q.sub.0], [mu] > 0 are constants defined in (21).

Remark 9. Let c > [bar.c] and N > 0; it follows from Lemmas 7 and 8 that the function

[[bar.[phi]].sub.N](x, y, t) := [PHI]([absolute value of (x)] - ct - M(1 - [e.sup.-[mu]t]) - N, y) + [q.sub.0][e.sup.- [mu]t] (59)

is a supersolution of (1) on t [member of] (0, +[infinity]).

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

Lemma 10. Let [[phi].bar](x, y, t) and [[bar.[phi]].sub.N](x, y, t) be defined by (42) and (59), respectively, and [u.sub.0](x, y) [member of] (0, 1) satisfies (38). Then, there exists N > 0 such that

[[phi].bar](x, y, t) [less than or equal to] u(x, y, t; [u.sub.0]) [less than or equal to] [[bar.[phi]].sub.N](x, y, t) (60)

for all (x, y, t) [member of] [R.sup.2] x [0, +[infinity]).

Proof. By (38) and the definition of (42), when t = 0, direct calculations give

[[phi].bar](x, y, 0) = [PHI](x, y) + [PHI](-x, y) - 1 - [q.sub.0] [less than or equal to] [PHI](x, y) [less than or equal to] [u.sub.0](x, y)

for x [greater than or equal to] 0. Similarly, we have [[phi].bar](x, y, 0) [less than or equal to] [u.sub.0](x, y) for x < 0. Therefore, the maximum principle for parabolic equations shows [[phi].bar](x, y, t) [less than or equal to] u(x, y, t; [u.sub.0]) for (x, y, t) [member of] [R.sup.2] x [0, +[infinity]).

Similarly, by the definition of (59), when t = 0

[[bar.[phi]].sub.N](x, y, 0) = [PHI]([absolute value of (x)] - N, y) + [q.sub.0]. (62)

By (38), we have that, for [epsilon] = [q.sub.0] and [N.sub.1] [greater than or equal to] 0, there exists [LAMBDA] > 0 such that

[mathematical expression not reproducible] (63)

in {(x, y) | [x.sup.2] + [y.sup.2] > [[LAMBDA].sup.2]}, since [[PHI].sub.z](z, y) < 0.

In the range of {(x, y) | [x.sup.2] + [y.sup.2] [less than or equal to] [[LAMBDA].sup.2]}, we have [absolute value of (x)] [less than or equal to] [LAMBDA]. Thus, by choosing [N.sup.2] [greater than or equal to] [LAMBDA] - M and using (20), (21), and (26), we have

[mathematical expression not reproducible] (64)

for (x, y) [member of] {(x, y) | [x.sup.2] + [y.sup.2] [less than or equal to] [[LAMBDA].sup.2]}. Consequently, we have [mathematical expression not reproducible] in this range.

Combining (63) and (64) and taking N [greater than or equal to] max{[N.sub.1], [N.sub.2]}, we conclude that [u.sub.0](x, y) [less than or equal to] [[bar.[phi]].sub.N](x, y, 0) for (x, y) [member of] [R.sup.2]. Then the maximum principle for parabolic equations derives that (60) holds.

Proof of Theorem 3. It follows from Lemma 10 that (60) holds with N [greater than or equal to] max{[N.sub.1], [N.sub.2]}. Thus, by (11), we obtain

[mathematical expression not reproducible] (65)

for all (x, y, t) [member of] [R.sup.2] x [0, +[infinity]), if [[zeta].sub.1] [greater than or equal to] M. On the other hand, for x [greater than or equal to] 0, we have

[mathematical expression not reproducible] (66)

if [[zeta].sub.2] [greater than or equal to] M + N and q [greater than or equal to] ([q.sub.0] + k). By a similar argument, for x < 0, (66) also holds.

Thus, if [u.sub.0](x, y) [member of] (0, 1) satisfies (38), by taking [zeta] = max{[[zeta].sub.1], [[zeta].sub.1]} and q [greater than or equal to] ([q.sub.0] + k), we obtain that (39) holds, where [q.sub.0] and [mu] > 0 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 [R.sup.2]. It means that the solution u(x, y, t, [u.sub.0]) of (1) with initial function [u.sub.0](x, y) 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 [R.sup.2], 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 3isa difficult problem, since the level set of the traveling curved fronts [PHI](z, y) of (1) have two asymptotic directions as [absolute value of (z)] [right arrow] +[infinity], and both directions make an angle with the negative y-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.

https://doi.org/ 10.1155/2017/5328246

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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Nai-Wei Liu (1,2)

(1) School of Mathematics, Shandong University, Jinan, Shandong 250100, China

(2) School 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 Koksal
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Date:Jan 1, 2017
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