GLOBAL STABILITY OF V-SHAPED TRAVELING FRONTS IN COMBUSTION AND DEGENERATE MONOSTABLE EQUATIONS

. This paper is concerned with the global stability of V-shaped traveling fronts in reaction-diﬀusion equations with combustion and degenerate monostable nonlinearity. The existence of such curved fronts has been recently proved by [39]. In this paper, by constructing new subsolutions, we show the asymptotic stability of V-shaped traveling fronts.

1. Introduction and main results. Parabolic differential equations can be used to model a host of natural processes such as biological invasions, combustion, chemical reactions, population dynamics, the spreading of diseases and others. An important class of solutions modeling the propagation of reaction is traveling wave front. In the past years, there had been many works devoted to the study of it, see [1,6,8,9,10,11,19,22,24,36] and many other works. These literatures mainly focused on planar traveling wave solutions in one-dimensional or high-dimensional spaces. In recent years, there were many studies on nonplanar traveling wave solutions such as V-shaped waves in two-dimensional spaces, pyramidal traveling waves in three-dimensional spaces and conical traveling waves in high-dimensional spaces. The investigation on these multi-dimensional traveling waves has important applications to multi-dimensional chemical waves, multi-dimensional curved flames and nerve transmission phenomena.
For the following equation with bistable nonlinearity, the existence and stability results of nonplanar traveling wave solutions were established by many authors in the dimensional N ≥ 2, see [12,15,16,20,26,27,31,32,33,34,35]. For the Fisher-KPP nonlinearity, Hamel and Nadirashvili [17] and Huang [18] proved that the equation (1) exists nonplanar traveling fronts and these fronts are stable with N ≥ 2. For the equation (1) with combustion nonlinearity, Bonnet and Hamel [2] established the existence of two-dimensional V-shaped traveling fronts with N = 2. Furthermore, Hamel and Monneau [13], investigated the questions of the existence, of the uniqueness and of the qualitative properties with conical shape in R N (N ≥ 2) and Hamel et al. [14] established the asymptotic stability of V-shaped traveling wave fronts in R 2 . Very recently, by constructing new supersolutions, the authors of this paper [5,39] established the existence and stability of V-shaped traveling waves in two-dimensional spaces and pyramidal traveling fronts in three-dimensional spaces for the equation (1) with combustion and degenerate Fisher-KPP nonlinearity. For more results about nonplanar traveling fronts of reaction-diffusion systems and periodic reaction-diffusion equations, we refer to [4,7,25,30,43,37,38,39,41,42].
In the one-dimensional space, a traveling wave front of (2) connecting the equilibrium u ≡ 0 and the equilibrium u ≡ 1 with speed k ∈ R means a function φ(x + kt) which satisfies where = x + kt. In this paper, we further assume that the following condition holds.
(A2) There exists a traveling front φ( ) ∈ C 2 (R) with speed c * > 0 satisfying (4) and where 0 and c * are two real roots of the equation λ 2 − c * λ = 0. Obviously, the assumptions (A1) and (A2) hold when the nonlinear term f is of combustion type. In this case, the solutions (c * , φ) of (4) are unique, in the sense that c * is unique and φ is also unique up to a translation in , see [1]. If the function f (u) = u p (1 − u) (p > 1), then it follows from [21] and the references therein that there exists a positive number c * such that the equation (4) has solutions (k, φ) if and only if k ≥ c * . Moreover, the traveling wave front φ( ) with minimal speed c * satisfies Thus, in this case, the assumptions (A1) and (A2) also hold and the number c * is just the minimal wave speed of one-dimensional traveling wave fronts of (4) connecting equilibria 0 and 1.
Under the conditions (A1) and (A2), we can obtain that there exist some positive constants K 1 , K 2 , K 3 , K 4 , K 5 and Λ such that Without loss of generality, we use the moving coordinate of speed c toward the -y direction. Put z = y + ct and u(t, We write the solution as v(t, x, z; v 0 ). If V is a traveling wave with speed c, it satisfies Throughout this paper, we always assume that c > c * . Let It is clear that the equation K(λ) = 0 has two roots 0 and c * 8 . In addition, we can easily obtain that K(c * β) < 0 if β ∈ 0, 1 8 . Note that φ c * c (z ± m * x) satisfy (9). They are so called planar traveling fronts. Since the maximum of subsolutions is also a subsolution, it turns out that is a subsolution of (9). V − (x, z) is strictly monotone increasing in z. In particular, the solution v(t, x, z; V − ) monotonely converges to a traveling curved front V (x, z) ( Figure 1), namely The following theorem has been proved in [39].
Moreover, one has For any β ∈ (0, 1), and 2254 ZHEN-HUI BU AND ZHI-CHENG WANG then the solution v(t, x, z; v 0 ) of (7)-(8) satisfies  Here we would like to give some comments on the traveling curved front V (x, z) in Theorem 1.1. We note that (10) implies that the curved front V (x, z) converges asymptotically to two planar traveling fronts φ c * c (z + m * |x|) along the two halflines z + m * |x| = const. and the level sets of the solution V have two asymptotic lines z + m * |x| = const.. See Figure 1 and Figure 2 for the level sets of the traveling curved front V . In fact, two asymptotic lines z + m * |x| = const. are the level set of . The level sets of V look like V-shaped curves in the x-z plane, so the curved front V (x, z) is also called a V-shaped front. We denote the angle between two asymptotic lines z + m * |x| = const. by 2α ∈ (0, π), which is also the angle of the V-shaped front V (x, z). Then we have α = arccotm * ∈ 0, π 2 , c = c * / sin α and φ c * c (z + m * |x|) = φ (z sin α + |x| cos α)). For the combustion nonlinearity, it is well known that (1) admits a unique planar wave front (c * , φ) satisfying (4), in the sense that c * is unique and φ is also unique up to a translation. But for V-shaped fronts of (1) in R 2 , the admissible wave speed is no longer unique, but a half continuum (c * , +∞). Following the formula c = c * / sin α, we know that the angle α is the smaller as the speed c is larger. The physical meaning is that the curvature (of the flame, eg.) increase with the speed (of the fuel flow, eg.), see [2]. Mathematically, the traveling curved front V (x, z) evolves from two oblique planar waves φ (z sin α + |x| cos α)) and describes their interaction. Thus, the propagation speed c of V (x, z) depends on the angle 2α between two oblique planar waves φ (z sin α + |x| cos α)).
Theorem 1.1 shows that the solution v(t, x, z; v 0 ) converges to the V-shaped traveling curved front when the initial value v 0 (x, z) ∈ C R 2 , [0, 1] is larger than the construction subsolution V − (x, z) and satisfies (12). However, by the continuity of the initial function, we can also expect that the solution of equation (7)-(8) converges to V (x, z) under the assumption (12) even if an initial value is less than V − (x, z). In this paper, our aim is to prove that the theorem holds true if the initial value only satisfies (12) without the assumption (13).
Remark 1. The stability of the traveling curved front V implies that for each c > c * , the solution (c, V ) of (9) and (10) is unique. Namely, if V (x, z) is another V-shaped traveling front of (9) and (10) with speed c, then one has In fact, for the reaction-diffusion equations with combustion nonlinearity, Hamel et al. [14] had proved the stability of the V-shaped traveling fronts in weighted spaces. In order to make a comparison with our result, we state the stability result in [14] specifically as follows.
Chooseα ∈ (0, π/2). Denote by U C R 2 the space of all bounded uniformly continuous functions from R 2 to R. We fix a C ∞ function h : R → R such that h (x) = |x| for |x| large enough. For ω 1 > 0, we set The space G ω1 is a Banach space with the norm In [14], Hamel et al. proved the following result.
Then there are four constants T ≥ 0, K ≥ 0, ω 1 > 0 and ω 2 > 0 such that Theorem 1.3 establishes the asymptotic stability of V-shaped traveling fronts of (9) if the nonlinearity f is only of combustion type and the initial value is less than the translation of the V-shaped traveling fronts. In contrast to Theorem 1.3, by constructing new subsolutions and using the comparison principle, which are motivated by Ninomiya and Taniguchi [26], in Theorem 1.2 we prove the asymptotic stability of V-shaped traveling fronts of (9) when the nonlinear terms f are of combustion and degenerate monostable type, and the initial value only satisfies (12) which is weaker than the assumptions in Theorem 1.3.
We will construct those subsolutions in Section 2 and give the proof of Theorem 1.2 in Section 3. Throughout this paper, we always assume that the conditions (A1) and (A2) hold. For the sake of convenience, in this paper we always let the function V (x, z) be the V-shaped traveling front established in Theorem 1.1. In addition, we always denote That is, the function c * − cψ (ζ) √ 1+ψ (ζ) 2 has a strict positive lower bound if ζ > 0. Thus the inequality (21) holds for all ζ ∈ R.
In the following we give at first the definition of mild super-and subsolutions of (7) and establish a comparison principle.
Lemma 2.3. Assume that (A1) hold. Then for any pair of mild supersolution w + (t, x, z) and mild subsolution w In the remainder of this paper, we always let b := c 8k1 . Now, we construct a mild subsolution to (9) as follows.
By the above arguments, we have proved (22). This completes the proof.
By (10) and Lemma 3.1, we obtain that there exists a constant R > 0 large enough such that for all (x, z) ∈ R 2 with x 2 + z 2 > R 2 . Then there exists a sufficiently large positive constant R 1 such that for all (x, z) ∈ R 2 with z + m * |x| < −R 1 . Thus, we have < v(T, x, z).