Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations

For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [4]. This paper gives another proof of the existence of axisymmetric traveling fronts. Our method is as follows. We use pyramidal traveling fronts for unbalanced reaction-diffusion equations, and take the balanced limit. Then we obtain axisymmetric traveling fronts in a balanced bistable reaction-diffusion equation. Since pyramidal traveling fronts have been studied in many equations or systems, our method might be applicable to study axisymmetric traveling fronts in these equations or systems.

1. Introduction. In this paper we study a reaction-diffusion equation where n ≥ 2 is a given integer, and given u 0 ∈ X. Here X is the set of bounded and uniformly continuous functions from R n to R with the norm The reaction term is called balanced when G(1) = G(−1) and is called unbalanced when G(1) = G(−1). When the reaction term is unbalanced multidimensional traveling fronts have been studied by [15,16,10,11,12,17,18,19,13,23,26,14,20,21,22] and so on. In this case, the propagation is mainly driven by the imbalance of the reaction kinetics and the curvature effect of an interface. Here a level set of a solution is often called an interface.
When the reaction term is balanced, one has no driven force caused by the reaction kinetics and the propagation is mainly driven by the curvature effect of an interface and is also driven by interaction between portions of an interface. For Equation (1.1), axisymmetric traveling fronts have been studied by Chen, Guo, Hamel, Ninomiya and Roquejoffre [4]. See del Pino, Kowalczyk and Wei [5] for a stationary solution, that is a traveling front with speed zero, related with De Giorgi's conjecture. See [6] for a traveling wave solution with two non-planar fronts and for a traveling wave solution with non-convex fronts. Recently the existence of axially asymmetric traveling fronts was studied by [24] as a balanced limit of pyramidal traveling fronts in unbalanced reaction-diffusion equations. See Wang [25] for traveling waves of a mean curvature flow in R n . In this paper we prove the existence of an axisymmetric traveling front solution to a balanced reaction-diffusion equation (1.1) by using the method of [24]. This axisymmetric traveling front solution is monotone decreasing in the traveling axis x n and travels with any given positive speed. Since pyramidal traveling fronts have been studied in many equations or systems as in [17,18,19,13,26,14,1,23], our method might be applicable to study axisymmetric traveling fronts in these equations or systems.
Let c > 0 be arbitrarily given. Let x = (x , x n ) ∈ R n with x = (x 1 , . . . , x n−1 ) ∈ R n−1 . We put z = x n − ct and u(x , x n , t) = w(x , z, t), and have Now we write z simply as x n . Then w(x, t) satisfies Then the profile equation for a traveling wave V with speed c is given by Let s * be the largest zero point of G in (−1, 1), that is, s * ∈ (−1, 1) is defined by We fix θ 0 with s * < θ 0 < 1 and have −G (θ 0 ) > 0. Let r = |x | and let 0 = (0, . . . , 0) ∈ R n−1 . In Section 4, we define for all (x , x n ) in any compact set in R n . Here V (mi) ki (x , x n ) is a pyramidal traveling front given by Theorem 2. See Figure 2 for its level set. As is seen in Section 4, we can define U (r, x n ) by Then we have The following is the main assertion in this paper.
Remark 1. From Theorem 1 the following assertion follows. For every θ ∈ (−1, 1), one has for any given compact set K in R 2 .
As far as the author knows, the uniqueness of an axisymmetric traveling front for (1.1) is yet to be studied. The author conjectures that an axisymmetric traveling front in Theorem 1 coincides with that of [4]. This is an open problem.
This paper is organized as follows. In Section 2, we make preparations. In Section 3, we show properties of pyramidal traveling fronts to unbalanced reactiondiffusion equations. In Section 4, we take the balanced limit of pyramidal traveling fronts, and prove Theorem 1.

Preliminaries. We extend
We put Following [16,4,17,18,19,24], we introduce a one-dimensional traveling front. For any k with Then we have Let k 0 ∈ 0, G (−1) be small enough such that one has Then we have Φ(0) = 0 and Thus Φ satisfies and is a one-dimensional traveling front with speed k ∈ (0, k 0 ). Now Φ also satisfies Thus Φ is a planar stationary front to (1.1).
See [7,2] for the proof of this uniqueness.
3. Properties of pyramidal traveling fronts to unbalanced reaction-diffusion equations. In this section we study properties of pyramidal traveling fronts for unbalanced reaction-diffusion equations. Two-dimensional V-form fronts and pyramidal traveling fronts in R n have been studied by [15,16,10,11,12,17,18,19,13,23,26,14] and so on. Let c > 0 be arbitrarily given. For a given bounded and uniformly continuous function u 0 let w(x, t; u 0 ) be the solution of For any k ∈ (0, min{k 0 , c}), let Let N be the set of positive integers and and letN = N ∪ {0}. For m ∈ N with m ≥ 2 we define J as For each j = (j 1 , . . . , j n−2 ) ∈ J, we define Here (a j , x ) denotes the inner product of vectors a j and x . In this paper we call We call the set of edges of a pyramid. For γ > 0, let Pyramidal traveling fronts are stated as follows. For the proof see [15] for n = 2 and see [17,13] for n ≥ 3.
Theorem 2 ( [15,17,13,23]). Let c > 0 be an arbitrarily given number. For every k ∈ (0, min{k 0 , c}), let h (m) and v be given in (3.2) and (3.4) Speed C Since h (m) is symmetric with respect to a plane (x , a j ) = 0, V (m) k ( · , x n ) is symmetric with respect to the same plane for any fixed x n ∈ R by the definition of V Then, using the definition of V (m) , we obtain for every j ∈ J. For every k ∈ (0, min{k 0 , c}) and every m ∈ N with m ≥ 2, we choose z 4. Balanced limits of pyramidal traveling fronts. In this section we study the limits of pyramidal traveling fronts for unbalanced reaction-diffusion equations as the reaction term approaches to a balanced one, and prove the existence of axisymmetric traveling fronts in balanced reaction-diffusion equations.
Let a sequence (k i ) i∈N satisfy lim i→∞ k i = 0 and We choose z i ∈ R with V for all (x , x n ) in any compact set in R n . Without loss of generality, we can assume that this convergence holds true for all i ∈ N. Then U 0 (x) satisfies the profile equation Since V (m) k ( · , x n ) is symmetric with respect to a plane (x , a j ) = 0 for any fixed x n ∈ R, U 0 ( · , x n ) is spherically symmetric in R n−1 for any fixed x n ∈ R. Using r = |x |, we define Now we have U (0, 0) = θ 0 . Using (3.6), we obtain ∂U ∂r (r, x n ) ≥ 0, r > 0, x n ∈ R. Now we have ∂ 2 U ∂r 2 (0, x n ) ≥ 0, x n ∈ R. (4.5) We will show the following lemma.

Lemma 1. One has
Proof. It suffices to prove the latter two inequalities. If ∂U/∂x n = 0 at some point in [0, ∞) × R, we have (∂U/∂x n )(r, x n ) ≡ 0 from the maximum principle. Then U (r, x n ) is independent of x n and is a function of r ≥ 0. By (1.4) and (4.5), we have Combining this equality and (4.5), we find −G (U (0, x n )) ≤ 0 for all x n ∈ R. This contradicts U (0, 0) = θ 0 and −G (θ 0 ) > 0. Thus we have ∂U ∂x n (r, x n ) < 0, r ≥ 0, x n ∈ R.
Next we prove the last inequality. Assume U r = 0 at some point (r 0 , x 0 n ) withr 0 > 0 and x 0 n ∈ R. Here we write U r (r, x n ) = (∂U/∂r)(r, x n ). Then U r satisfies Then, using the maximum principle, we get U r ≡ 0 on [0, ∞) × R. Then U (r, x n ) is independent of r ≥ 0 and is a function of x n , and satisfies Since the one-dimensional traveling front profile is uniquely determined and its speed is uniquely determined by [2], we obtain c = 0 and U (r, x n ) = Φ(x n ) for all x n ∈ R. This contradicts c > 0. Thus we obtain the last inequality. This completes the proof. Now the following assertion follows from [24].
Proof. The proof of this proposition can be carried out by a simplified argument as in the proof of Theorem 1 of [24].
Remark 2. For every θ ∈ (−1, 1), a level set {x ∈ R n | U 0 (x) = θ} is given by a graph of a function that is defined on the entire space R n−1 .
Now we have the following lemma.

Lemma 3.
There exists a positive constant m 0 such that the following assertions hold true. For every x ∈ Ω − one has For every x ∈ Ω + one has Proof. For any given L > 0, we define Then P satisfies and

Now we have
For every y ∈ Ω + , let p(y) be given by √ np(y) = dist(y, ∂Ω + ), where dist(y, ∂Ω + ) is the distance from y to ∂Ω + . Now u(x) = u 0 (x) − P (x − y; p(y)) satisfies we have Then we obtain from the maximum principle. Especially we have Then the former half of the lemma follows from the Schauder interior estimate of [8]. The latter half can be proved similarly. Now we show the level set of U (r, x n ) becomes parallel to the x n -axis as r → ∞. We prove this lemma in Section 5. The following lemma asserts the asymptotic behavior of U .
for any given compact set K in R 2 .
Proof. Let r 0 > 1 be given and we define for (x, y) ∈ R 2 with

MASAHARU TANIGUCHI
Using Lemma 1, we have Using Lemma 4, we see that is a function of x and is independent of y. Using (1.4) and Proposition 1, we have Then we obtain u 1 (x) = Φ(x + a) with a ∈ R and Φ(a) = θ. This completes the proof. Now Theorem 1 follows from Lemma 1, Proposition 1, Lemma 4 and Lemma 5.
See [8] for instance. We define γ θ (x) by v(x, γ θ (x)) = θ for every x ∈ R. Then γ θ (x) is of class C 1 (R). We define The, by using Lemma 3, there exist constants m 1 > 0 and µ > 0 such that we have In the following, we will show a contradiction following Gui [9]. We introduce x (x, y)v y (x, y) dy for any fixed x ∈ R.