GLOBAL ASYMPTOTIC STABILITY OF TRAVELING WAVES TO THE ALLEN-CAHN EQUATION WITH A FRACTIONAL LAPLACIAN

. In this paper, we study the asymptotic stability of traveling wave fronts to the Allen-Cahn equation with a fractional Laplacian. The main tools that we used are super- and subsolutions and squeezing methods.

In the past few years, many researchers paid their attention to the traveling waves for reaction-diffusion equations with a fractional Laplacian, see [5,15,1,16,17]. Mellet et al. [15] considered a fractional reaction-diffusion equation with a combustion nonlinearity, in which the sliding argument was used to establish the existence of traveling waves of the problem where the fractional power s ∈ ( 1 2 , 1). Gui and Huan [10] proved the nonexistence of traveling wave fronts for fractional reaction-diffusion equation with a combustion nonlinearity with s ∈ (0, 1 2 ]. Gui and Huan [10] also proved the existence and nonexistence of traveling wave fronts for the generalized Fisher-KPP model with the fractional Laplacian. For the bistable type model, Gui and Zhao [11] got the existence and uniqueness of monotonically increasing traveling wave fronts for this problem, where they focused on the unbalanced case (namely, 1 0 f (u)du = 0) and s ∈ (0, 1). The main tool they used is the continuation method. Other related works on this topic have been considered in [3,4,5,6,7] and references therein.
For the classical Allen-Cahn equation, its traveling wave fronts have been widely studied. Fife and Mcleod [9] studied the global exponential stability of one-dimensional traveling waves. Ninomiya and Taniguchi [18] studied the existence and the stability of V-shaped traveling wave fronts in two-dimensional space. Taniguchi [21,22] proved the existence, uniqueness and asymptotic stability of pyramidal traveling fronts in three-dimensional space. Matano et al. [14] studied the stability of planar waves in high-dimensional space. There were many follow-up studies for this problem.
We also note that Chen [8] studied the existence, uniqueness and global exponential stability of traveling wave fronts of a class of nonlinear and nonlocal evolution equations. Smith and Zhao [20] established the global exponential stability of traveling wave fronts in delayed reaction diffusion equations. Wang et al. [23] established the existence, uniqueness and global exponential stability of traveling wave fronts in reaction diffusion equations with nonlocal delays. Their methods give us a lot of inspiration. To the best of our knowledge, there is no result for the global asymptotic stability of traveling wave fronts for the Allen-Cahn equation with a fractional Laplacian. The aim of this paper is to present the global exponential stability of traveling waves of the problem (1) under the assumption (H).
In this paper, we care about the one dimension case, which means x ∈ R. Traveling wave fronts of (1) are the solutions with the form u(x, t) = U (x − ct), where c is called the speed of the traveling waves. When the assumption (H) holds, it follows from [11] that (1) admits a unique traveling wave front (U, c) satisfying Furthermore, by [11], the traveling wave front U of (1) has the following properties: In the next section, we first build the comparison principle for (1) and introduce a very important inequality. Then we establish several pairs of the super-and subsolutions. In the last section, we will use the squeezing method to prove our main result. Our main result reveals the global exponential stability of traveling wave fronts. The methods we used are based on [8,20]
By the definition, it is observed that }. Now we have the following lemma.
where h(x, t) ∈ C([0, T ), X). Then The lemma can be proved by applying the standard theory of the strongly continuous semigroup [19,24], so we omit the details.

LUYI MA, HONG-TAO NIU AND ZHI-CHENG WANG
Clearly, if a function u(x, t) = u(t)(x) ∈ C([0, T ), X) is both a mild supersolution and a mild subsolution, then it is a mild solution.
Due to Lemma 2.1, it is easy to see that if a function u(x, t) is a classical supersolution (subsolution) of (7) on t ∈ [0, T ), then it must be a mild supersolution (subsolution) of (7) on t ∈ [0, T ).
The following theorem gives the existence of solutions, the comparison principle and a key inequality.
In addition, assume that u + (x, t) and u − (x, t) are a mild supersolution and a mild subsolution of (1) on [0, ∞) respectively, and satisfy u |f (u)| and B > 1 is defined by (A3) above.
Consequently, we get The proof is complete.
In the following we establish two pairs of super-and subsolutions of (1) on [0, ∞), which is very important to prove our main result in next section. Lemma 2.5. Assume that (H) holds. Let (U, c) be the traveling wave front of (1) satisfying (3) and (H1). Then there exist positive numbers β 0 , σ 0 andδ such that for every δ ∈ (0,δ] and ξ 0 ∈ R, the functions u + and u − defined by are a classical supersolution and a classical subsolution of (1) on [0, ∞), respectively.
We consider it in three cases.

STABILITY OF TRAVELING WAVES TO FRACTIONAL LAPLACIAN EQUATION 2465
Case I: ρ(ε(x − Ct)) < δ 2 . In this case, we have −δ ≤ z − (x, t) < − δ 2 . By (11) and (13), we get that (12) and (14), we have In summary, we get Then z − (x, t) is a subsolution of (1) on [0, ∞). We can prove z + (x, t) is a supersolution of (1) on [0, ∞) in a similar way. This completes the proof. Remark 1. It is worth remarking that z + (x, t) and z − (x, t) in Lemma 2.6 have the following properties: 3. Stability of traveling wave front. In this section, we establish the global asymptotic stability of traveling wave front for (1). In order to prove the main result, we first give the following two lemmas. Throughout this section, let U (x−ct) be the traveling wave front of (1) satisfying (3) and (H1). By Lemma 2.5, we define the following two functions: for x ∈ R, t ∈ [0, ∞), ξ ∈ R and δ ∈ (0,δ), whereδ, σ 0 , β 0 are as in Lemma 2.5.

STABILITY OF TRAVELING WAVES TO FRACTIONAL LAPLACIAN EQUATION 2467
Thus, By the mean value theorem and the choice of N 0 and J, we know that, for all |x − z| ≥ J, Then, for all |x − z| ≥ J, Therefore, let t = 1 in (16), we have for all |x − z| ≥ J. By (17) and (18), we obtain that, for all x ∈ R, That is, By Theorem 2.4, for all t ≥ 0, we obtain For any t ≥ T + 1, let t = t − (T + 1) in (19), then whereδ(t) = (δe −β0 + ε * h )e −β0(t−(T +1)) . By the choice of η and the monotonicity of U (·), we have It is easy to see thatξ .