# Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations inR2

Authors: Nai-Wei Liu
Journal: Journal of Mathematics
 (2017)

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## 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 inR2 under appropriate initial conditions.


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## Body

## 1. Introduction

In the current paper, we consider the following Cauchy problem:(1)∂u∂t=Δu+fu,x,y∈R2,t>0,ux,y,0=u0x,y,x,y∈R2,where u0∈C1(R2) is a bounded initial function and the function f is of bistable type. Concretely, we assume that f satisfies the following:(F1)
f∈C1(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)
∫01‍f(s)ds>0.Such an example is (2)fu=u1-uu-a,for0<a<12.Under assumptions (F1) and (F2), it is clear that there exists a positive constant δ(0<δ<1/2) such that (3)f′u≤-γ1,u∈-∞,2δ∪1-2δ,∞,where γ1≔1/2min⁡{-f′(0),-f′(1)}>0.We remark here that the steady states0 and 1 of (1) are asymptotically stable if (F1)–(F3) hold. In R1, by letting u(x-c-t)=ϕ(ξ) and ξ=x-c-t(c->0), one has (4)-ϕ′′ξ-c-ϕ′ξ-fϕ=0,ϕ′ξ<0,ξ∈R,limξ→-∞ϕξ=1,limξ→∞ϕξ=0.It is well known that (4) has a planar traveling wave front ϕ(ξ) which is unique up to phase shift under assumptions (F1)–(F3), with the unique positive traveling wave speed 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 x-direction and let (5)ux,y,t=Vz,y,t,z=x-ct;then, we can rewrite (1) into (6)Vt-Vzz-Vyy-cVz-fV=0,z,y∈R2,t>0,Vt=0=u0,z,y∈R2.For convenience, we denote the solution of Cauchy problem (6) with initial function V(z,y,0;u0)=u0(z,y) by V(z,y,t;u0).Considering the traveling wave frontsu(x,y,t)=Φ(z,y)(z=x-ct) of (1) with traveling wave speed c in x-direction, then (7)-Φzz-Φyy-cΦz-fΦ=0,inR2.It is obvious that the solution of (7) is a stationary solution of (6). Let c≥c-; then ϕc-/cz±m∗y is a solution of (7), where ϕ(·) is a solution of (4) and (8)m∗≔c2-c-2c->0.We call the solution Φ(z,y) of (7)traveling curved front, since it is nonplanar. By using comparison principle, one has the function (9)φz,y≔max⁡ϕc-cz-m∗y,ϕc-cz+m∗y=ϕc-cz-m∗ywhich is a subsolution of (7) with φz(z,y)<0 on R2. 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 anyc>c-, there exists a traveling curved front u(x,y,t)=Φ(z,y)(z=x-ct) of (1) such that (10)limR→+∞supz2+y2>R2Φz,y-φz,y=0,(11)Φz,y>φz,y,0<Φz,y<1,Φzz,y<0,forz,y∈R2.Furthermore, if 0<δ1<1/2, there is a constant γ(δ1)>0 such that (12)Φzz,y≤-γδ1,forδ1≤Φ≤1-δ1.If u0(z,y) satisfies (13)limR→+∞supz2+y2>R2u0z,y-Φz,y=0,then (14)limt→+∞supx,y∈R2ux,y,t;u0-Φx-ct,y=0,where u(x,y,t;u0) is the solution of the Cauchy problem (1).It follows from Theorem1 that (1) has a unique traveling curved front Φ(x-ct,y) for each c>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 R3. 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→+∞. 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 R1, 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-calledentire solutions. Entire solutions can be used to imply the dynamics of two traveling fronts as t→-∞; one can refer to [33–37] for related works.However, the interaction of traveling curved fronts of reaction-diffusion equations in whole spacesR2 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 R2 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 R2.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 frontΦ(z,y) of (1) as z→-∞ by using the result of the exponential convergence of one-dimension traveling wave solution ϕ(ξ) of (4) at infinity. In fact, it follows from [38] that there exist positive constants A and B such that (15)ϕξ=Ae-μ~ξ+oe-μ~ξ,asξ⟶+∞,1-ϕξ=Beλ~ξ+oeλ~ξ,asξ⟶-∞,where μ~=c-2-4f′(0)+c-/2 and λ~=c-2-4f′(1)-c-/2. From [34], we see that the planar traveling wave front ϕ of (4) satisfies (16)1-ϕξ≤keλ~ξ,ξ≤0for some k>0 and λ~ defined above.Under conditions (F1)–(F3) and (3), there exists a constant γ2 with γ2>γ1>0, such that (17)fu≤γ21-ufor 0<1-u≤δ with δ as in (3). Furthermore, by virtue of (12), we have (18)Φzz,y≤-γδ≔-γ3<0for δ≤Φ≤1-δ. Since the traveling wave front ϕξ of (4) possesses invariance up to translation, we assume that traveling wave front ϕξ satisfies (19)ϕ0=θ,θ≥1-δ2,and the constant k in (16) satisfies (20)k≤min⁡γ1δ4γ2-γ1,δ4.We take three positive constants q0, μ, and M satisfying (21)δ4≤q0≤δ2,0<μ≤min⁡γ12,λc,M≥max⁡M1,M2,where (22)M1≔kL2+γ2+L2+μq0γ3μ,M2≔L3+μq0γ3μ,L2≔max-δ≤u≤1-δf′u,L3≔max-δ≤u≤1+δf′u.By a translation in the ξ-direction, we next take (23)ϕ~ξ=ϕξ-M,where M is defined in (21). Then, we have (24)1-ϕ~ξ+M=1-ϕξ≤keλ~ξ,ξ≤0,(25)ϕ~M=ϕ0=θ,θ≥1-δ2,by view of (16) and (19).In the following, we consider planar traveling wave frontϕ~(ξ) satisfying (24) and (25) instead of the solution ϕ(ξ) of (4) and assume that Theorem 1 holds with ϕ~(ξ) instead of ϕ(ξ) in the definition of φ(z,y). For convenience, in the rest of the paper we drop the tilde of ϕ~ and denote ϕ~(ξ) also by ϕ(ξ).By using the asymptotic behavior of planar traveling wave fronts of (4), we immediately obtain the following lemma.Lemma 2.
Assume thatf satisfies (F1)–(F3). Then there exist some positive constants λ, k, and C, such that the traveling curved front Φ(z,y) defined in Theorem 1 satisfies (26)0≤1-Φz+M,y≤keλz-m∗y,z≤0,y∈R,(27)Φzz,y≤C,z,y∈R2,where M is defined in (21) and (28)λ≔c-cλ~=c-c-2-4f′1-c-22c>0.Furthermore, there is (29)limR→+∞supz-m∗y>RΦzz,y=0.Proof.
It follows from (9), (11), and (23) that (30)0≤1-Φz+M,y≤1-φz+M,y=1-ϕc-cz+M-m∗y≤1-ϕc-cz-m∗y+M.Thus (26) holds for z≤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 ε1>0 and {(zn,yn)}n=1∞ satisfying (31)limn→∞zn+m∗yn=∞,Φzzn,yn≥ε1.Define (32)Φnz,y=Φz+zn,y+yn,inB0,where (33)B0≔z,y∈R2∣z2+y2<C,with C>0 a given constant. By extracting subsequence of Φn and denoting the subsequence also by Φn, we have (34)Φnz,y⟶Φ∗z,y,inCloc2R2,where Φ∗(z,y) is a solution of (7). On the other hand, by view of (10) and (11), we have (35)limn→∞supB0¯Φnz,y-1=0,limn→∞supB0¯fΦnz,y=0.Thus the strong maximum principle implies (36)∂∂zΦ∗z,y=0,inB0.Then, (37)∂∂zΦnz,y⟶0,asn⟶∞,which contradicts the assumption Φz(zn,yn)≥ε1>0. Thus, we complete the proof.Our main result is the following.Theorem 3.
For everyc>c-, let Φ(·,y) be the traveling curved front of (1) defined in Theorem 1 with speed c. Assume that (F1)–(F3) hold. Then if u0(x,y)∈(0,1) satisfies (38)u0x,y≥Φx,y,limR→+∞supx2+y2>R2u0x,y-Φx,y=0,there exist positive constants q,μ>0 and ζ, such that, for all (x,y,t)∈R2×[0,+∞), the solution u(x,y,t;u0) of (1) satisfies (39)Φx-ct+ζ,y+Φ-x-ct+ζ,y-1-qe-μt≤ux,y,t;u0≤Φx-ct-ζ,y+Φ-x-ct-ζ,y-1+qe-μt.Furthermore, one has (40)limt→+∞ux,y,t;u0-1=0locally uniformly with respect to (x,y)∈R2.Remark 4.
Inequality (39) implies that the x-profile of u(x,y,t;u0) 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 ζ 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 R2×(0,+∞).Definition 5.
If a functionu_(x,y,t)∈C2,1(R2×(0,+∞),R) and satisfies (41)∂u_x,y,t∂t≤Δu_x,y,t+fu_x,y,t,x,y∈R2,t>0,then u_(x,y,t) is called a subsolution for (1) in R2×(0,+∞). Similarly, by reversing the inequality in (41), we can define a supersolution u-(x,y,t) for (1).


## 3. Proof of Theorem3

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. Letc>c-. Then the function (42)ϕ_x,y,t≔Φx-ct+M1-e-μt,y+Φ-x-ct+M1-e-μt,y-1-q0e-μtis a subsolution of (1) on t∈(0,∞), where Φ(·,y) is traveling curved front of (1) defined in Theorem 1 and q0,μ>0 are constants defined in (21).Proof.
Define(43)Fϕ_x,y,t≔ϕ_x,y,t∂t-Δϕ_x,y,t-fϕ_x,y,t.By using the above prepared results, direct calculations give (44)Fϕ_=Mμe-μt-cΦzx-ct+M1-e-μt,y+Mμe-μt-cΦz-x-ct+M1-e-μt,y+q0μe-μt-Φzzx-ct+M1-e-μt,y-Φzz-x-ct+M1-e-μt,y-Φyyx-ct+M1-e-μt,y-Φyy-x-ct+M1-e-μt,y-fΦx-ct+M1-e-μt,y+Φ-x-ct+M1-e-μt,y-1-q0e-μt=Mμe-μtΦzx-ct+M1-e-μt,y+Mμe-μtΦz-x-ct+M1-e-μt,y+q0μe-μt+fΦx-ct+M1-e-μt,y+fΦ-x-ct+M1-e-μt,y-fΦx-ct+M1-e-μt,y+Φ-x-ct+M1-e-μt,y-1-q0e-μt.
Ifx≥0, we consider two cases Φ(x-ct+M1-e-μt,y)∈[0,δ]∪[1-δ,1] and Φ(x-ct+M1-e-μt,y)∈[δ,1-δ], respectively.
Case
A (
Φ
(
x
-
c
t
+
M
1
-
e
-
μ
t
,
y
)
∈
[
1
-
δ
,
1
]
). By virtue of (3), (21), and (25), we have (45)fΦx-ct+M1-e-μt,y-fΦx-ct+M1-e-μt,y+Φ-x-ct+M1-e-μt,y-1-q0e-μt≤-γ11-Φ-x-ct+M1-e-μt,y+q0e-μt,since 0<1-Φ-x-ct+M1-e-μt,y+q0e-μt≤1-ϕ(M)+q0≤δ. By using (17) and (25) and the fact that 0<1-Φ-x-ct+M1-e-μt,y≤1-ϕ(M)≤δ/2 for x≥0, we have (46)fΦ-x-ct+M1-e-μt,y≤γ21-Φ-x-ct+M1-e-μt,y.Thus, we have (47)Fϕ_≤-γ11-Φ-x-ct+M1-e-μt,y+q0e-μt+γ21-Φ-x-ct+M1-e-μt,y+q0μe-μt≤γ2-γ11-Φ-x-ct+M1-e-μt,y-γ1-μq0e-μt≤kγ2-γ1eλ-x-ct-m∗y-γ1-μq0e-μt≤kγ2-γ1e-λct-γ1-μq0e-μt≤0.In the last inequality, we have used the facts (20), (21), and (26).
By a similar argument, we haveF(ϕ_)≤0 for Φ(x-ct+M1-e-μt,y)∈[0,δ] with x≥0.
Case
B (
Φ
(
x
-
c
t
+
M
1
-
e
-
μ
t
,
y
)
∈
[
δ
,
1
-
δ
]
). In a similar way as above, for x≥0, we have (48)fΦx-ct+M1-e-μt,y-fΦx-ct+M1-e-μt,y+Φ-x-ct+M1-e-μt,y-1-q0e-μt≤L21-Φ-x-ct+M1-e-μt,y+q0e-μt,where L2≔max-δ≤u≤1-δf′(u). Particularly, we have (46) in this case. Lastly, due to (18), we have (49)Φz′x-ct+M1-e-μt,y+Φz′-x-ct+M1-e-μt,y≤-γ3.Combining (46), (48), and (49), we have (50)Fϕ_≤-γ3Mμe-μt+L21-Φ-x-ct+M1-e-μt,y+q0e-μt+γ21-Φ-x-ct+M1-e-μt,y+q0μe-μt≤-γ3Mμe-μt+L2+γ21-Φ-x-ct+M1-e-μt,y+L2+μq0e-μt≤-γ3Mμe-μt+kL2+γ2e-λct+L2+μq0e-μt.Consequently, (21) implies F(ϕ_)≤0 for this case.
Similarly, we can proveF(ϕ_)≤0 when x<0. Thus, we have showed that ϕ_(x,y,t) is a subsolution of (1) on t∈[0,∞).In order to construct a supersolution for (1), we introduce the following lemmas.Lemma 7.
Assume that (F1)–(F3) hold. Letc>c-; then for any given constant N>0, the function (51)ϕN1x,y,t≔Φx-ct-M1-e-μt-N,y+q0e-μtis a supersolution of (1) on t∈(0,+∞), where Φ(·,y) is traveling curved front of (1) as in Theorem 1 and q0,μ>0 are constants defined in (21).Proof.
As the proof of Lemma6, we need only to prove that the right hand of (43) is nonnegative for the function ϕN1(x,y,t) for (x,y,t)∈R2×(0,+∞). By a similar argument, direct calculations give (52)FϕN1=-Mμe-μt-cΦzx-ct-M1-e-μt-N,y-q0μe-μt-Φzzx-ct-M1-e-μt-N,y-Φyyx-ct-M1-e-μt-N,y-fΦx-ct-M1-e-μt-N,y+q0e-μt=-Mμe-μtΦzx-ct-M1-e-μt-N,y-q0μe-μt+fΦx-ct-M1-e-μt-N,y-fΦx-ct-M1-e-μt-N,y+q0e-μt.
To complete the proof, we consider two casesΦ(x-ct-M1-e-μt-N,y)∈[0,δ]∪[1-δ,1] and Φ(x-ct-M1-e-μt-N,y)∈[δ,1-δ], respectively.
ForΦ(x-ct-M1-e-μt-N,y)∈[0,δ]∪[1-δ,1], we just consider the case Φ(x-ct-M1-e-μt-N,y)∈[1-δ,1]. Since Φx-ct-M1-e-μt-N,y+q0e-μt∈[1-δ,1+δ], it follows from (3) that (53)fΦx-ct-M1-e-μt-N,y+q0e-μt-fΦx-ct-M1-e-μt-N,y≤-γ1q0e-μt.By virtue of (11) and (21), we have (54)FϕN1≥-Mμe-μtΦzx-ct-M1-e-μt-N,y-q0μe-μt+γ1q0e-μt≥q0γ1-μe-μt≥0.
ForΦ(x-ct-M1-e-μt-N,y)∈[δ,1-δ], we have (55)fΦx-ct-M1-e-μt-N,y-fΦx-ct-M1-e-μt-N,y+q0e-μt≥-L3q0e-μt,where L3≔max-δ≤u≤1+δf′(u). Particularly, (18) implies (56)Φz′x-ct-M1-e-μt-N,y≤-γ3.Therefore, by (21) we get (57)FϕN1≥γ3Mμe-μt-q0μe-μt-L3q0e-μt≥γ3Mμe-μt-L3+μq0e-μt≥0.
Thus,ϕN1(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. Letc>c-; then for any given constant N>0, the function (58)ϕN2x,y,t≔Φ-x-ct-M1-e-μt-N,y+q0e-μtis a supersolution of (1) on t∈(0,+∞), where Φ(·,y) is traveling curved front of (1) as in Theorem 1 and q0,μ>0 are constants defined in (21).Remark 9.
Letc>c- and N>0; it follows from Lemmas 7 and 8 that the function (59)ϕ-Nx,y,t≔Φx-ct-M1-e-μt-N,y+q0e-μtis a supersolution of (1) on t∈(0,+∞).To complete the proof of Theorem3, we establish the following comparison result.Lemma 10.
Letϕ_(x,y,t) and ϕ-N(x,y,t) be defined by (42) and (59), respectively, and u0(x,y)∈(0,1) satisfies (38). Then, there exists N>0 such that (60)ϕ_x,y,t≤ux,y,t;u0≤ϕ-Nx,y,tfor all (x,y,t)∈R2×[0,+∞).Proof.
By (38) and the definition of (42), when t=0, direct calculations give (61)ϕ_x,y,0=Φx,y+Φ-x,y-1-q0≤Φx,y≤u0x,yfor x≥0. Similarly, we have ϕ_(x,y,0)≤u0(x,y) for x<0. Therefore, the maximum principle for parabolic equations shows ϕ_(x,y,t)≤u(x,y,t;u0) for (x,y,t)∈R2×[0,+∞).
Similarly, by the definition of (59), when t=0(62)ϕ-Nx,y,0=Φx-N,y+q0.By (38), we have that, for ε=q0 and N1≥0, there exists Λ>0 such that (63)u0x,y≤Φx,y+q0≤Φx-N1,y+q0≤ϕ-N1x,y,0in {(x,y)∣x2+y2>Λ2}, since Φz(z,y)<0.
In the range of{(x,y)∣x2+y2≤Λ2}, we have |x|≤Λ. Thus, by choosing N2≥Λ-M and using (20), (21), and (26), we have (64)ϕ-N2x,y,0≥1-keλx-M-N2-m∗y+q0≥1-keλΛ-M-Λ-M-m∗y+q0≥1-k+q0≥1for (x,y)∈{(x,y)∣x2+y2≤Λ2}. Consequently, we have ϕ-N2(x,y,0)≥u0(x,y)∈(0,1) in this range.
Combining (63) and (64) and taking N≥max⁡{N1,N2}, we conclude that u0(x,y)≤ϕ-N(x,y,0) for (x,y)∈R2. Then the maximum principle for parabolic equations derives that (60) holds.Proof of Theorem3.
It follows from Lemma10 that (60) holds with N≥max⁡{N1,N2}. Thus, by (11), we obtain (65)ϕ_x,y,t=Φx-ct+M1-e-μt,y+Φ-x-ct+M1-e-μt,y-1-q0e-μt≥Φx-ct+ζ1,y+Φ-x-ct+ζ1,y-1-q0e-μtfor all (x,y,t)∈R2×[0,+∞), if ζ1≥M. On the other hand, for x≥0, we have (66)ϕ-Nx,y,t=Φx-ct-M1-e-μt-N,y+q0e-μt≤Φx-ct-M1-e-μt-N,y+Φ-x-ct-M1-e-μt-N,y-1+q0e-μt+1-Φ-x-ct-M1-e-μt-N,y≤Φx-ct-M1-e-μt-N,y+Φ-x-ct-M1-e-μt-N,y-1+q0e-μt+keλ-x-ct-M-N-m∗y≤Φx-ct-M1-e-μt-N,y+Φ-x-ct-M1-e-μt-N,y-1+q0e-μt+ke-λct≤Φx-ct-M1-e-μt-N,y+Φ-x-ct-M1-e-μt-N,y-1+q0+ke-μt≤Φx-ct-ζ2,y+Φ-x-ct-ζ2,y-1+qe-μtif ζ2≥M+N and q≥(q0+k). By a similar argument, for x<0, (66) also holds.
Thus, ifu0(x,y)∈(0,1) satisfies (38), by taking ζ=max⁡{ζ1,ζ1} and q≥(q0+k), we obtain that (39) holds, where q0 and μ>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 inR2. It means that the solution u(x,y,t,u0) of (1) with initial function u0(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 R2, 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 Theorem3 is a difficult problem, since the level set of the traveling curved fronts Φ(z,y) of (1) have two asymptotic directions as z→+∞, 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.



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