Multi-dimensional traveling fronts in bistable reaction-diffusion equations

This paper studies traveling front solutions of convex polyhedral shapes 
in bistable reaction-diffusion equations including the 
Allen-Cahn equations or the Nagumo equations. 
By taking the limits of such solutions as the lateral faces go to 
infinity, we construct 
a three-dimensional traveling front solution for 
any given $g\in C^{\infty}(S^{1})$ with 
$\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.

1. Introduction. In this paper we consider the following equation A given function u 0 belongs to BU (R 3 ). Here BU (R 3 ) is the space of bounded uniformly continuous functions from R 3 to R with the supremum norm. The Laplacian ∆ stands for ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 . Here f is a bistable or multi-stable nonlinear term. Here we write a typical example: f (u) = −(u+1)(u+a)(u−1) with a ∈ (0, 1). This equation is called the Allen-Cahn equation, the Nagumo equation or the scalar Ginzburg-Landau equation. It appears in various fields, say, in population genetics [10], ecology [35,34], bistable transmission in electronic circuits [25], phase transitions in metallurgy, van der Waals theory and Landau theory [1,21] and chemical reactions [40]. Traveling fronts are often observed in these fields. For the literature for one-dimensional traveling fronts, one can refer to Kanel' [18,19], Nagumo, Yoshizawa and Arimoto [25], Aronson and Weinberger [2,3], Fife and McLeod [9] and X. Chen [6] and many other works.
We show f in Example 1 or Example 2 satisfies (A1), (A2) and (A3) in Lemma 2.1. We derive the profile equation for a traveling wave with speed c. We adopt the moving coordinate of speed c toward the z-axis without loss of generality. We put s = z − ct and u(x, y, z, t) = w(x, y, s, t). We denote w(x, y, s, t) by w(x, y, z, t) for simplicity. Then we obtain Here w t stands for ∂w/∂t and so on. We write the solution as w(x, y, z, t; u 0 ). If v is a traveling wave with speed c, it satisfies Equation (3) is the profile equation for traveling waves.
Here we briefly explain well-known traveling waves for bistable reaction-diffusion equations. For stationary ball solutions, see [4] and [29] and references therein. Recently del Pino, Kowalczyk and Wei [8] studied stationary solutions related to de Giorgi conjecture.
Multi-dimensional traveling front solutions have been studied recently. Twodimensional V-form fronts are studied by Ninomiya and myself [27,28] and Hamel, Monneau and Roquejoffre [13,14]. See also Haragus and Scheel [17]. Traveling fronts with cylindrical symmetry are studied by [13,14,7]. Pyramidal traveling fronts are studied by [37,38] in the three-dimensional space and by Kurokawa and myself [22] in the N -dimensional space for N ≥ 4.
The aim of this paper is as follows. 1. Traveling front solutions of convex polyhedral shapes are constructed in Theorem 3.2 of §3. We call them generalized pyramidal traveling front solutions. 2. For any given g ∈ C ∞ (S 1 ) with min 0≤θ≤2π g(θ) = 0, we construct a threedimensional traveling front solution in Theorem 6.1 by sending the number of lateral faces of a convex polyhedron to infinity in §5. Let n be as in (7). The cross section of a convex polyhedron {(x, y, z) ∈ R 3 | z ≥ h(x, y)} for sufficiently large z > 0 is an n-polygon. Here h is given by (7). To guarantee the convergence of generalized pyramidal traveling front solutions in Theorem 3.2 to three-dimensional traveling front solutions in Theorem 6.1, we need a uniform estimate on the width of generalized pyramidal traveling front solutions with respect to n. We discuss this uniform estimate in §4 and prove Theorem 6.1 in §5.

Preliminaries.
Hereafter we assume c > k.
Since the curvature often accelerates the speed, we have many traveling front solutions when c > k. Though it is an interesting problem to study the traveling front solutions when c ≤ k, we just study c > k in this paper.
We set where Θ is a number given by (17). We note that Ω j = ∅ for all 1 ≤ j ≤ n.
which is a part of an edge of a polyhedron {(x, y, z) ∈ R 3 | z ≥ h(x, y)}. If (c 1 , . . . , c n ) = (0, . . . , 0), Θ = 0 is valid and Γ j and ∪ n j=1 Γ j represent an edge and the set of all edges for a pyramid, respectively.
Equation (3) has a solution Φ ((k/c)(z − h j (x, y))) . It is called a planar traveling front associated with the lateral face S j . Now we put v(x, y, z) We define for γ ≥ 0. We state the uniqueness and stability of a two-dimensional V-form front in the two-dimensional plane.
Thus a pyramidal traveling front solution V is asymptotically stable globally in space if a given fluctuation decays at infinity. Now we call V as in Theorem 2.3 the pyramidal traveling wave V (x; p) associated with a pyramid z = p(x, y).
We set and define A pyramidal traveling front is uniquely determined as a combination of twodimensional V-form fronts.
Lemma 2.5. Let p(x, y) be given by (13) and let V be as in Theorem 2.3. If Proof. We have Using both sides as initial values of (2) and taking the limit of t = +∞, we obtain due to Theorem 2.3. This completes the proof.
3. Generalized pyramidal traveling fronts. In this section we study traveling front solutions of convex polyhedral shapes. In this section we assume only (A1) and (A2). We first note that Since h j is affine we have for all 1 ≤ j ≤ n and any θ ∈ (0, 1). Thus we have For any ζ ∈ R and 1 ≤ j ≤ n, let (X j (ζ), Y j (ζ)) be defined by that is,

MASAHARU TANIGUCHI
For every ζ ∈ R a set {(x, y) ∈ R 2 | h(x, y) ≤ ζ} is the empty set or a non-empty convex closed set in R 2 . Indeed, for i = 1, 2 let (x i , y i ) satisfy h(x i , y i ) ≤ ζ. Then we have h j (x i , y i ) ≤ ζ for all 1 ≤ j ≤ n and i = 1, 2. Since h j is affine, we have On the other hand, it becomes a convex n-polygon if ζ > 0 is large enough. We find and fix such a large number. For this purpose we set and fix any positive number ρ ∈ (Θ, ∞).
, Y n (ρ)) are located counterclockwise and a set {(x, y) | h(x, y) ≤ τ ρ} becomes an n-polygon. The intersection point of the two lines is Thus, if 0 < θ j+1 − θ j−1 < π, ρ should satisfy which is equivalent to We have Now we can see that ρ > Θ gives (18). This completes the proof.
We have h(X j (ρ), Y j (ρ)) = τ ρ for all 1 ≤ j ≤ n. Thus we obtain Let V 0 (x) denote the pyramidal traveling front V (x; p) as in Theorem 2.3. Let v be as in (8). Then v is a subsolution to (3) and satisfies v(x, y, z) is a supersolution to (3). We define for all (x, y, z) ∈ R 3 . See Sattinger [33, Theorem 3.6] for general arguments.
Using (12), (15) and lim R→∞ sup |x|≥R |V (x) − max 1≤j≤n E j (x)| = 0 and applying the Schauder interior estimate [11,Theorem 9.11 For all 1 ≤ j ≤ n we have Then we get We consider the left-hand side or the right-hand side as an initial value of (2), respectively. Using Theorem 2.2 and sending t → +∞, we obtain for all (x, y, z) ∈ R 3 . We set Considering the left-hand side or the right-hand side as an initial condition of (2) and sending t → ∞, we obtain Theorem 3.2 (Generalized pyramidal traveling front solutions). Let c > k and let h(x, y) be given by (7). Under the assumptions (A1) and (A2) there exists a solution V (x, y, z) to (3) with Moreover one has Then lim holds true. Especially V (x, y, z) is uniquely determined by (3) and (21).
Proof. It suffices to prove the uniqueness and stability of V . We have and define for every x ∈ R 3 . Then V * satisfies We choose a positive constant σ with Similarly as in [27,Lemma 4.4] we find that is a supersolution to (2) for any δ ∈ (0, δ * ). For any given δ ∈ (0, δ * ), we take λ > 0 large enough and get Then we have Sending t → ∞, we obtain We define We have Λ ≥ 0. If Λ = 0, V * ≡ V follows. We prove Λ = 0 by contradiction. Assume Λ > 0. Then the comparison principle gives Now we fix R 0 > 0 large enough and get 2σ sup Taking Combining these two estimates together, we obtain Then we get Sending t → +∞, we obtain This contradicts the definition of Λ. Thus V * ≡ V follows. This implies Applying a similar argument to For any fixed t ≥ 0, w(x, t; · ) is a continuous mapping in BU (R 3 ). Using this continuity, Theorem 2.3 and the comparison principle, we obtain This completes the proof. (7). We call it a generalized pyramid and call V in Theorem 3.2 the generalized pyramidal traveling front solution associated with z = h(x, y), and denote it by V (x; h). See Figure 2. For the cross section of V (x; h) see Figure 3.
Let h(x, y) be as in (7) and let h(x, y) be Using each side as an initial value of (2) and taking the limit of t = +∞, we obtain As in Theorem 2.3, V is strictly monotone decreasing in z. This suggests that V is strictly monotone decreasing along a direction near the z-axis. Indeed we have the following lemma.
be a given constant vector with Then one has and have Then we get Thus we get x ∈ R 3 , ε ∈ (0, 1).

By Theorem 3.2 we have
Sending t → +∞, we have Applying the strong maximum principle to This completes the proof.
We now consider the case where h is symmetric with respect to the x-axis and the y-axis, respectively.
Then one has Proof. First V (x, y, z) = V (|x|, |y|, z) follows from the uniqueness in Theorem 3.2.
We have The comparison principle gives For any ε > 0 we have Sending t → +∞, we have Sending ε → 0, we have V x ≥ 0 if x > 0. Then the strong maximum principle gives Similarly we have This completes the proof.

4.
Uniform estimates on widths of generalized pyramidal traveling front solutions. In this section we deal with generalized pyramidal traveling front solutions in Theorem 3.2, and give estimates of widths of their internal transition layers uniformly in n, where n is the number of lateral faces. For this purpose we use expanding balls, that is, developing fronts with spherical symmetry, and give upper and lower estimates of V by the comparison principles. Hereafter we assume (A1), (A2) and (A3). First we give an upper bound of V . Let ψ(0) = α in (A3). Let From Theorem 3.2 and Lemma 3.4 we have We consider the following boundary value problem: Then the maximum principle gives for all x ∈ Ω 0 , t > 0.
Now we construct a supersolution that gives an upper estimate on V . For given Q > 0, δ > 0 and D > 0, let q(t) be given by Taking Q > 0 larger and taking δ > 0 smaller if necessary, we can assume Then we have q (t) > 0 for all t > 0. We fix a constant λ 1 > 0 to satisfy sup µ∈R e λ1|µ| |ψ (µ)| < ∞, and we put pending only on f the following holds true. If Q > 0 is large enough and δ > 0 is small enough, w(r, t) given by (25) satisfies Proof. The same proof in [42, Propositions 3.1] can be carried out under the condition (A1), (A2) and (A3). If L > 0 is small enough, we have for all 0 ≤ r ≤ L and t > 0, since Q is large enough. This gives the last inequality. If Especially the convergence lim z→+∞ V (x 0 , y 0 , z 0 + z) = −1 is uniform in n and h and x 0 .
Next we give lower estimates on V uniformly in n. Let V be as in Theorem 3.2 and let x 0 = (x 0 , y 0 , z 0 ) satisfy V (x 0 , y 0 , z 0 ) = α. We put We denote w(x, t; α) simply by w(t; α), since it depends only on t. We have Then we obtain Here recall that for all x > 0.
We define a positive constant as Especially one has Proof. The argument stated above gave the first inequality. It suffices to prove the second inequality. Since we have Putting t = bz, we get This completes the proof.
By using x ≥ x 0 and y ≥ x 0 and z ≤ z 0 from Lemma 3.5 and Theorem 3.2. We put D 2 def = {(x, y, z) ∈ R 3 | x > x 0 and y > x 0 and z < z 0 }.
Proof. We have Then we complete the proof by the argument stated above.
Combining an upper estimate and a lower estimate, we obtain the following estimate on V .
for all z > 0. Here constants Q > 0 and δ > 0 are as in Lemma 4.1.
Proof. This proposition follows from Lemma 4.2 and Lemma 4.3.
Let x 0 = (x 0 , y 0 , z 0 ) satisfy V (x 0 ) = α. This proposition implies that the convergence is uniform in n and h and x 0 .

5.
The limits of pyramidal traveling fronts as the number of lateral faces goes to infinity. Traveling fronts with cylindrical symmetry are studied in [13] and [14] for bistable nonlinearity.
Here we construct cylindrically symmetric traveling front solutions by another method under the assumptions (A1), (A2) and (A3). We consider pyramids whose cross sections are regular 2 n -polygons. In the limit of n → ∞ we study traveling front solutions with cylindrical symmetry. For every n ≥ 2 we set and put x cos 2πj 2 n + y sin Let V (n) be the pyramidal traveling front solution associated with z = p (n) (x, y) as in Theorem 2.3. We define z Then we have U (n) (0, 0, 0) = 0, For any compact set K 1 ⊂ R 3 and p 1 > 3, (U (n) ) n∈N is bounded in L p1 (K 1 ). The Schauder interior estimates implies that (U (n) ) n∈N is bounded in W 2,p1 (K 1 ). The compact imbedding W 2,p1 (K 1 ) ⊂ C 1,α1 (K 1 ) for α 1 < 1 − 3/p 1 implies that (U (n) ) n∈N has a subsequence that converges in C 1,α1 (K 1 ). Let U be the limit of the subsequence of (U (n) ) n∈N . Then using the Schauder interior again, a subsequence of (U (n) ) n∈N converges to U in C 2 (K 1 ). By the diagonal argument we see that a subsequence of (U (n) ) n∈N converges to U in C 2 loc (R 3 ). We have Now V (n) and U (n) are symmetric with respect to a plane (x, y, z) | y cos πj 2 n = x sin πj 2 n for each 0 ≤ j ≤ 2 n − 1. Thus U has the same symmetry, which implies that U has cylindrical symmetry with respect to the z-axis, that is, a function of r = x 2 + y 2 and z. We write U = U (r, z) for r ≥ 0 and z ∈ R. Then it satisfies U r (0, z) = 0 for z ∈ R, U (0, 0) = 0.
Proof. Since U (n) satisfies Proposition 1, we have estimates on the limiting function U by sending n → ∞. This completes the proof.
This lemma implies that the convergence is uniform in (r 0 , z 0 ).
Proof. It suffices to prove lim inf r→∞ φ (r) > 0. Then the latter statement follows from Lemma 5.1 and Lemma 5.2. Assume the contrary. Then there exists (r i ) i∈N ⊂ (0, ∞) with lim i→∞ r i = ∞ and lim i→∞ φ (r i ) = 0. Then we have lim i→∞ U r (r i , φ(r i )) = 0. For any given a > 0 we take i with r i > 2a and have Then the Harnack inequality [11, Corollary 9.25] gives 0 ≤ sup B((ri,φ(ri));a) where a constant C is independent of i. Taking a subsequence of r i and call the subsequence (r i ) if necessary, we have ψ 0 (z) def = lim i→∞ U (r + r i , z + φ(r i )). We note that the right-hand side is independent of r.
We have Combining this equality and Lemma 5.1, we have which contradicts the uniqueness of a one-dimensional traveling front solution in [6] or [9], because we have c > k and we already have a one-dimensional traveling front solution (k, Φ). This completes the proof. Proof. For any r > 1 we set v(x, z; r) and have Sending r → +∞ and using Lemma 5.1, we have a limit function We have We put We obtain a contradiction by assuming τ < τ . We set c ∈ (0, c) by We have φ(|x|) ≤ c |x| for all r ≥ 0.
Since the absolute value of the gradient of This contradicts v 0 (0, 0) = 0 by sending t → +∞.
If f is of a bistable type, a cylindrically symmetric traveling front U coincides to that of [13] and [14] due to [14,Theorem 3]. For the uniqueness of U for a bistable nonlinear term f , see [14]. The uniqueness of U for a multistable nonlinear term f is yet to be studied. 6. Limits of generalized pyramidal traveling fronts. We set We identify S 1 with R/2πZ. Let g ∈ C ∞ (S 1 ) be any given function with min 0≤θ≤2π g(θ) = 0.
Speed C x y z Figure 6. The graph of U .
This contradicts the definition of Λ 1 . Thus U ≡ U follows. This completes the proof.
Thus a convex and bounded domain D with a smooth boundary C on a twodimensional plane gives a three-dimensional traveling front U in the Allen-Cahn equation. If another smooth convex closed curve C satisfying the conditions stated above is given, we have U (x; C) ≡ U (x; C ) from the theorem. See Figure 5. The cross section of {x ∈ R 3 | U (x) = 0} at z converges to that of a graph z = φ(r−R(θ)) (0 ≤ θ ≤ 2π) as z → +∞, that is, we have lim z→+∞ dist({(x, y) ∈ R 2 | U (x, y, z) = 0}, {(r cos θ, r sin θ) ∈ R 2 | z = φ(r −R(θ))}) = 0.
The cross section of {x ∈ R 3 | U (x) = 0} at z is almost a circle after rescaling if z > 0 is large enough, that is, converges to a unit circle as z → +∞.

7.
Conjecture. If f is of bistable type and satisfies the assumptions of [13,14], U in § 5 coincides with the cylindrically symmetric traveling front solution in [14,Theorem 2] due to [14,Theorem 3]. See also [26]. Then [14] gives φ(r) = τ r − k 0 log r + O(1) as r → ∞ with a constant k 0 > 0. Then we have This implies that z n 0 in § 5 satisfies lim n→∞ = ∞ and that V (n) converges to −1 in C 2 loc (R 3 ) as n → ∞. This is because the curvature effect makes V (n) go upwards as n → ∞.
The zero level of U is given by z = φ(r). The radius of the cross section of z = φ(r) at z ∈ R is given by r = φ −1 (z). We can see that φ −1 (z) is faster than linear growth. On the other hand, the zero level surface of a pyramidal traveling front converges to a pyramid. Thus the cross section at z ∈ R is a polygon and the scale is proportional to z as z → ∞. One might suspect that the difference of the growth rate of cross sections implies that there exists no traveling front whose cross section is the mixture of a circle and a polygon as in See Figure 7. The existence or non-existence of such mixed traveling front is yet to be studied.
8. Discussion. Generalized pyramidal traveling fronts or traveling fronts of various kinds of smooth shapes are constructed in this paper for a bistable or multistable reaction-diffusion with unbalanced energy potentials. For balanced bistable or multistable f , that is, For multi-dimensional traveling fronts for reaction-diffusion equations of combustion types or Fisher-KPP types, one can see [12] and [15] for instance. In Smith and Pickering [36] or Buckmaster [5], flames of polyhedral shapes or flames of various kinds of smooth shapes are observed for burning gas in Bunsen burners. From these experiments one can suspects that reaction-diffusion equations of combustion types or Fisher-KPP types or reaction-diffusion systems might have traveling fronts of convex polyhedral shapes or various kinds of smooth shapes. This problem is also interesting and should be studied in future.