PERIODIC TRAVELING WAVES OF A MEAN ...

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 25, Number 1, September 2009

doi:10.3934/dcds.2009.25.231 pp. 231–249

PERIODIC TRAVELING WAVES OF A MEAN CURVATURE FLOW IN HETEROGENEOUS MEDIA

Bendong Lou Department of Mathematics, Tongji University Shanghai 200092, China

Dedicated to Professor Masayasu Mimura. Abstract. We consider a curvature flow in heterogeneous media in the plane: V = a(x, y)κ + b, where for a plane curve, V denotes its normal velocity, κ denotes its curvature, b is a constant and a(x, y) is a positive function, periodic in y. We study periodic traveling waves which travel in y-direction with given average speed c ≥ 0. Four different types of traveling waves are given, whose profiles are straight lines, “V”-like curves, cup-like curves and cap-like curves, respectively. We also show that, as (b, c) → (0, 0), the profiles of the traveling waves converge to straight lines. These results are connected with spatially heterogeneous version of Bernshte˘ın’s Problem and De Giorgi’s Conjecture, which are proposed at last.

1. Introduction. Consider the following equation V = a(x, y)κ + b

for

(x, y) ∈ Γt ⊂ R2 ,

(1)

where Γt is a simple plane curve, V denotes the normal velocity on Γt , κ is the curvature, b is a constant, and a(x, y) is a positive function (further conditions on a will be given below). Equation (1) is an example of mean curvature flows in heterogeneous media. Mean curvature flows appear not only in geometry (cf. [8] and references therein), but also in singular perturbation problems for reaction diffusion equations ([7], [10], [14], [17], etc.) modeling phenomena in chemistry, ecology, etc.. For example, in [14] we considered a singular limit problem 1 Ut = ∇(A(x)∇U) + 2 F (x, U), x ∈ Rn , t > 0, (2) ε where ε > 0 is a small parameter, and F is the derivative of a double-well potential (the well-depth are different in ε order). Taking ε → 0 we derived a mean curvature flow from (2), which describes the motion of the level set of U: V = (n − 1)a(x)κ + c(x)∇d(x) · n + b(x)

for

x ∈ Γt

(3)

where, Γt := {x | U(x, t) = C} is a hypersurface (also called interface) in Rn , n is the unit normal vector to Γt , V is the normal velocity and κ is the mean 2000 Mathematics Subject Classification. Primary: 35B27, 53C44; Secondary: 35B40. Key words and phrases. Mean curvature flow, periodic traveling wave, curved front, heterogeneous media. This work is supported by Innovation Program of Shanghai Municipal Education Commission (09ZZ33) and by NNSF of China (10671143).

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BENDONG LOU

curvature of Γt , and a, b, c, d are some smooth functions. In [8] and references therein, the authors considered equation (3) for n = 2. Among others, they studied the asymptotic behavior of Γt starting from a simple closed curve. In this paper, we are concerned with traveling waves of (1) (a special case of (3)). In the last two decades, many authors studied traveling waves of reaction diffusion equations like (2) in heterogeneous media (cf. [4], [15], [23] and references therein). Most of the studies are concerned with traveling waves with planar-like front (that is, each level set of the front lies in a bounded neighborhood of a hyperplane). Recently, [12], [13], [20] etc. studied traveling waves with curved fronts, but for homogeneous equations. As far as we know, not much is known about traveling waves with curved fronts for heterogeneous equations like (2). On the other hand, comparing with reaction diffusion equations, very little is known about the traveling waves of mean curvature flows. Since the dimension of the interface Γt is n − 1, smaller than the dimension of x, if one studies (3) instead of (2), more information about the interface such as the shape, the relation between the traveling speed and the shape are easier to be known. Especially, in the case n = 2, the interface reduces to a simple plane curve, and its shape can be described clearly, as we will see in this paper. In [5], [6], [16], [18], [19] etc., the authors studied traveling waves of homogeneous case of (1) V = aκ + b, (4) that is, the case a(x, y) ≡ a > 0 in (1). In what follows, we say a curve Γ is a graphic one if it is the graph of some function. We say a traveling wave is a graphic one if its profile is a graphic curve. In this paper, we first give a review about all possible graphic traveling waves of (4), and then use them to construct periodic traveling waves of (1). If, for each t > 0, Γt is the graph of y = u(x, t), then equation (1) is equivalent to p uxx ut = a(x, u) + b 1 + u2x . (5) 1 + u2x In this paper, we assume that a(x, y) is bounded, smooth, L-periodic in y and 0 < a∗ :=

inf

(x,y)∈R2

a(x, y)
0, we call a solution u(x, t) of (5) as a periodic traveling wave with average speed c if u (x, t + L/c) = u(x, t) + L. (7) For simplicity, a stationary solution u(x, t) ≡ u¯(x) of (5) is also called a periodic traveling wave with speed 0. Our main purpose in this paper is to study the existence of periodic traveling waves of (5), as well as the relation among the traveling speed, the shape of the profile and the spatially heterogeneity. First of all, in the case b > 0, p u(x, t) = ±x c2 − b2 /b + ct, x, t ∈ R

(for each c ≥ b) are traveling waves of (5) with planar profiles. Besides these trivial solutions, we will also present other types of traveling waves: “V”-like, cup-like and cap-like periodic traveling waves (see the definitions in Section 3). The following is one of our main results.

PERIODIC TRAVELING WAVES

233

Main Theorem. Let c > b > 0. Then equation (5) has a periodic traveling wave U (x, t) ∈ C 2+ν,1+ν/2 (R2 ) (ν ∈ (0, 1)), whose graph is a “V”-like curve for each t ∈ R. More precisely, ϕ(x, a∗ ) ≤ U (x, t) − ct ≤ ϕ(x, a∗ ) + (a∗ − a∗ )S

for x ∈ R,

(8)

where S = S(b, c) > 0 is a constant (see (13) below), ϕ(x, a) is defined by parameter ψ as the following: Z ψ Z ψ √c2 − b2 √c2 − b2 dψ ψ · dψ x(ψ) = , ϕ(ψ) = , for ψ ∈ − , , Φ(ψ) b b 0 Φ(ψ) 0 p where Φ(ψ) = (1 + ψ 2 )(c − b 1 + ψ 2 )/a.

Detailed definitions of S and ϕ are given in√Section 2. Among others, both ϕ(x, a∗ ) and ϕ(x, a∗ ) + (a∗ − a∗ )S approach ±x c2 − b2 /b − a∗ S as x → ±∞. We also study the asymptotic shapes of the periodic traveling waves as (b, c) → (0, 0), and give an extension of Bernshte˘ın’s Problem and De Giorgi’s Conjecture. Roughly speaking, Bernshte˘ın’s Problem states that, if the coefficients in (3) are constants and b ≡ 0. Then the stationary solution Γt of (3) is a hyperplane when Γt is the graph of a function defined on Rn−1 . De Giorgi’s Conjecture states that, if we consider the spatially homogeneous version of equation (2) with a cubic F , then the level sets of a stationary solution U of (2) must be hyperplanes. They were proved to be true for n ≤ 8 and false for n > 8. In this paper we will show that, when (b, c) → (0, 0), the profiles of various types of our traveling waves approach horizontal straight lines. This fact indicates that Bernshte˘ın’s Problem and De Giorgi’s Conjecture can be extended to spatially heterogeneous case: At least for n ≤ 8, if the coefficients in (2) and (3) are periodic (resp. almost periodic), then the level sets of stationary solutions of (2) and the stationary interfaces of (3) are quasi-periodic (resp. almost periodic) (see details in Section 5). This paper is organized as follows. In Section 2, we summarize all types of graphic traveling waves of (4). In Section 3, we first prove the existence and uniqueness of “V”-like periodic traveling waves in band domains, then extend them to the whole plane. In Section 4, we study cup-like and cap-like periodic traveling waves in band domains. In Section 5, we extend Bernshte˘ın’s Problem and De Giorgi’s Conjecture to spatially heterogeneous cases. 2. Traveling waves in homogeneous media. As preliminaries, we first consider graphic traveling waves of (4). Most of the results in this section have been mentioned in [16], [18], [19]. For reader’s convenience, we give the details below. For graphic curves, (4) is equivalent to (5) with a(x, y) ≡ a > 0. A traveling wave of which is a solution with the form u(x, t) = ϕ(x, a) + ct (c ≥ 0). Here ϕ(x, a) solves p ϕxx c=a + b 1 + ϕ2x . (9) 2 1 + ϕx

The graph of ϕ is called the profile of traveling wave ϕ(x, a) + ct. It is easily seen that, if ϕ(x, a) is a solution of (9), then ϕ(±x + C1 , a) + C2 (for any C1 , C2 ∈ R) are solutions too. Hence we normalize ϕ by ϕ(0, a) = 0 and ϕx (0, a) = 0, provided ϕx vanishes at some points. Set ϕx (x, a) = ψ(x, a), then (9) is rewritten as p 1 + ψ2 ψx = c − b 1 + ψ 2 =: Φ(ψ). (10) a

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To seek for a solution of (9), we only need to solve (10). According to the signs of b and c − |b|, we divide the parameter domain {(b, c)|b ∈ R, c ≥ 0} into the following 10 parts (see Figure 1): Case Case Case Case

1. c = b = 0; Case 2. c > b = 0; 4. c = b > 0; Case 5. b > c > 0; 7. c = 0 > b; Case 8. − b > c > 0; 10. c > −b > 0.

Case 3. c > b > 0; Case 6. b > c = 0; Case 9. − b = c > 0;

Cases 1, 6 and 7 are trivial cases where the curves remain stationary. Case 1. ϕ(x, a) = Cx for any C ∈ R since ψx (x, a) ≡ 0. Case 6 (see Figure 5). The profile is the upper half-circle with radius ab : h a a i a 2 a2 a x2 + ϕ(x, a) + = 2, ϕ(x, a) ≥ − x∈ − , . b b b b b a : Case 7 (see Figure 6). The profile is the lower half-circle with radius |b| h i a2 a a a a 2 x2 + ϕ(x, a) + = 2, ϕ(x, a) ≤ − x∈ ,− . b b b b b For other cases, c > 0, and so dψ ψ · dψ dx = , dϕ = ψdx = when Φ(ψ) 6= 0. Φ(ψ) Φ(ψ) Thus ϕ(x, a) is defined as the following Z ψ Z dψ a(s − s0 ) ab s ds x(ψ) − x(ψ0 ) = = − Φ(ψ) c c b − c cos s ψ0 s0   q  s tan s + c−b    as ab 2 c+b   ,   q log + √ Cases 2, 3, 10;   2 2 c s c−b  c c −b  tan 2 − c+b   s0  h as a  s i s  + cot , Case 4; = c c 2 s0 " !# r s    2a|b| b+c s as   √ arctan − tan , Cases 5, 8;   2 − c2 c b − c 2  c b  s0  h as a i s   s   − tan , Case 9, c c 2 s0 where ψ0 and s0 = arctan ψ0 are 0 except for Case 4, ψ ∈ R and s = arctan ψ. Moreover, p p Z ψ b 1 + ψ02 − c ψ · dψ a 1 + ψ2 ϕ(ψ) − ϕ(ψ0 ) = = log p · p . b 1 + ψ2 − c c 1 + ψ02 ψ0 Φ(ψ)

A tedious but direct calculation gives the following facts (see also [16], [18], [19]). Case 2 (see Figure 2). ϕ is even, convex and exists on (−xa , xa ) with xa = aπ 2c , ϕx (xa − 0, a) = +∞, ϕ(xa − 0, a) = +∞. Such a graph is called the Grim Reaper by some authors. Moreover, ϕx (x, a) is decreasing in a. So for a∗ > a∗ > 0, we have xa∗ > xa∗ and ϕx (x, a∗ ) < ϕx (x, a∗ ),

ϕ(x, a∗ ) < ϕ(x, a∗ )

Case 3 (see Figures 3a, 3b). (9) has 5 solutions. √ c2 −b2 (1) Straight line ϕ(x, a) = x; √b 2 2 (2) Straight line ϕ(x, a) = − c b−b x;

for x ∈ (0, xa∗ ).


235

(3) Increasing concave curve. For any given x1 , ϕ is defined on [x1 , +∞) and √ c2 − b 2 ϕx (x1 + 0, a) = +∞, ϕx (+∞, a) = , ϕxx (x, a) < 0 (x > x1 ); b (4) Decreasing concave curve. For any given x2 , ϕ is defined on (−∞, x2 ] and √ c2 − b 2 ϕx (x2 − 0, a) = −∞, ϕx (−∞, a) = − , ϕxx (x, a) < 0 (x < x2 ); b (5) “V”-shaped curve, namely, the graph of ϕ is like the letter “V”, which is even, convex, ϕ(0, a) = 0, and √ √ 2 2 c2 − b 2 − c cab−b |x| as x → ±∞, (11) ϕx (x, a) = ± +O e b √ √ c c2 −b2 c2 − b 2 ϕ(x, a) = |x| − aS + O |x|e− ab |x| b where

as x → ±∞,

√ √ c2 − b 2 c2 − b 2 1 2(c + b) S= arctan + log . cb b c b

(12)

(13)

Furthermore, ϕx (x, a) is decreasing in a. For a∗ > a∗ , ϕx (x, a∗ ) < ϕx (x, a∗ ) and ϕ(x, a∗ ) < ϕ(x, a∗ ),

ϕ(x, a∗ ) + a∗ S < ϕ(x, a∗ ) + a∗ S

for x 6= 0.

(14)

Case 4 (see Figure 4). (9) has 3 solutions. (1) Horizontal line ϕ(x, a) ≡ 0; (2) Increasing concave curve. For any given x1 , ϕ is defined on [x1 , +∞) and ϕx (x1 + 0, a) = +∞, ϕx (+∞, a) = 0, ϕ(+∞, a) = +∞, ϕxx (x, a) < 0 (x > x1 ); (3) Decreasing concave curve. For any given x2 , ϕ is defined on (−∞, x2 ] and ϕx (x2 − 0, a) = −∞, ϕx (−∞, a) = 0, ϕ(−∞, a) = +∞, ϕxx (x, a) < 0 (x < x2 ). Case 5 (see Figure 5). ϕ is even, concave and exists only on (−xa , xa ) where aπ ab xa = − + 2c c

Z

π 2

0

ds , b − c cos s

and ϕx (±xa , a) = ∓∞, ϕ(±xa , a) = ac log b−c b . Moreover, ϕx (x, a) is increasing in a. We call the graph of ϕ a cap-shaped curve. Cases 8-10 (see Figure 6). ϕ is even, convex and exists on (−xa , xa ) where aπ ab + 2c c

Z

π 2

ds , c cos s − b 0 and ϕx (±xa , a) = ±∞, ϕ(±xa , a) = ac log b−c b . The last equality is different from Case 2 where ϕ is unbounded. Moreover, ϕx (x, a) is decreasing in a. We call the graph of ϕ a cup-shaped curve. xa =

236

BENDONG LOU

c y

Case 2 Case 4

Case 9 Case 10

Case 3

Case 8

j(x,a *)

Case 5

j(x,a *) Case 7

b

Case 6

Case 1

-xa*

Fig. 1 parameter domain

-xa*

xa *

x

xa *

Fig. 2 Grim Reaper

y

y (5)

(1)

(2)

j(x,a *) + a*S j(x,a *) + a*S j(x,a *)

j(x,a *) (4)

x1

x2

(3) x 0

Fig. 3a Case 3 -- 5 types of travelling waves

y

(3) x2

x1

x

Fig. 3b Case 3 -- V-shaped curves

(2) (1) x

Fig. 4 Case 4

We are concerned with four types of curves. Type I: planar curves, i.e., straight lines ((1) and (2) in Case 3, (1) in Case 4); Type II: “V”-shaped curves ((5) in Case 3); Type III: cup-shaped curves (Cases 2,7,8,9,10); Type IV: cap-shaped curves (Cases 5,6). We call their corresponding traveling waves as planar, “V”-shaped, cup-shaped and cap-shaped traveling waves, respectively. 3. “V”-like periodic traveling waves. In this section we prove the Main Theorem. More specifically, for any given c with c > b > 0, we construct a periodic traveling wave of (5) which travels in y-direction with average speed c. Moreover, for each t ∈ R, the graph of the traveling wave is a “V”-like curve, which means that the graph lies between two “V”-shaped curves. Similarly, we call a periodic


237

traveling wave a cup-like one (resp. a cap-like one) if, for each t, the graph of which lies between two cup-shaped (resp. cap-shaped) curves. 3.1. “V”-like periodic traveling waves in band domains. 3.1.1. Setting of the initial-boundary value problems. To construct “V”-like profiles spanning on the whole R, we first consider equation (5) on finite intervals x ∈ [−h, h]. For this purpose, some boundary conditions are needed. In the study of curvature-driven motion of plane curves with endpoints, the contact angles at the endpoints like ux (−h, t) = B− (u(−h, t)),

ux (h, t) = B+ (u(h, t)),

t > 0,

(15)

are usually imposed, where B± are periodic or constant functions (cf. [2, 3, 16]). But in this paper, we will impose a Dirichlet boundary condition: u(±h, t) = ct + ϕ(h, a∗ ),

t > 0,

(16)

where ϕ(x, a∗ ) is that in (14). There are several points which indicate that the Dirichlet boundary condition is reasonable in our problem. First, the Dirichlet boundary condition does have physical meaning in some sense. Recall that, for a(x, y) in Section 1, we do not impose periodic or other conditions on x, so it is possible that a is a constant outside a band |x| < h1 . In this case, the problem outside the band is indeed spatially homogeneous, and so when h ≫ h1 , the motion of the curve outside the band is nearly a traveling wave which satisfies (16). Second, the main purpose of this paper is to give a “V”-like periodic traveling wave whose profile spans on the whole R. The problems on finite intervals is just our first stage, what we are really interested is the limiting profile as h → ∞. So the boundary conditions on finite intervals may leave influence to the limit, but it will not appear explicitly in the limiting problem. Third, the Dirichlet boundary conditions are of benefit to the description of the shape of the profile (see details below and Remark 4). Thus, we study the following problem. ( p uxx ut = a(x, u) + b 1 + u2x , x ∈ (−h, h), t > 0, 2 1 + ux (17) u(±h, t) = ct + ϕ(h, a∗ ), t > 0. To give a periodic traveling wave of (17) we need a time-global solution (see details in subsections 3.1.4-3.1.6), so we first consider (17) with initial data u(x, 0) = u0 (x),

x ∈ [−h, h].

(18)

Since we are interested in periodic traveling waves rather than the solvability of (17) for general initial data, we choose a special initial data u0 as the following: u0 ∈ C 3 [−h, h], and ϕ(x, a∗ ) ≤ u0 (x) ≤ ϕ(x, a∗ )+Dh ,

|u0x (x)| ≤ K,

|u0xx (x)| ≤ K

for x ∈ [−h, h]. (19) Here, Dh is the unique constant in (0, S ∗ ) ≡ (0, (a∗ − a∗ )S) satisfying ϕ(±h, a∗ ) = ϕ(±h, a∗ ) + Dh ,

(this is possible by (11)-(14)). K is a constant satisfying n o K > max sup |ϕx (x, a∗ )|, sup |ϕxx (x, a∗ )| . x∈R

x∈R

(20)

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BENDONG LOU

Note that |ϕx (x, a∗ )|, |ϕxx (x, a∗ )| < K for all x, note also that ϕ(x, a∗ ) is a special candidate for u0 (x). 3.1.2. Uniform H¨ older estimates. By a classical solution of (17)-(18) on time-interval [0, T ], we mean a solution in space C 2+ν,1+ν/2 (QT ) for some ν ∈ (0, 1), where QT = (−h, h) × (0, T ). In this part, we always assume that u(x, t) is a classical solution of (17)-(18) on time interval [0, T ], and then give some a priori estimates. Lemma 3.1. (see Figure 7). The following conclusions hold. (i) ϕ(x, a∗ ) + ct + Dh (resp. ϕ(x, a∗ ) + ct) is an upper solution (resp. lower solution) of (17)-(18), and so ϕ(x, a∗ ) ≤ u(x, t)−ct ≤ ϕ(x, a∗ )+Dh for (x, t) ∈ QT . (ii) ϕx (h, a∗ ) ≤ ux (h, t) ≤ ϕx (h, a∗ ), ϕx (−h, a∗ ) ≤ ux (−h, t) ≤ ϕx (−h, a∗ ) for t ∈ [0, T ]. (iii) kux kC(QT ) ≤ K. Proof. (i). The results follows from comparison principle. (ii). The results follows from (i) and (20). (iii). Set w := ux , then  a(x, u) bwwx   wx +√ , x ∈ (−h, h), t > 0,  wt = 2 1+w 1 + w2 x w(±h, t) = ux (±h, t), t > 0,    w(x, 0) = u0x (x), x ∈ [−h, h].

(21)

So conclusion (iii) follows from maximum principle and (ii). Define

(22) v(x, t) := u(x, t) − ct − ϕ(x, a∗ ) for (x, t) ∈ QT . We will give various estimates for u by studying v. First, Lemma 3.1 implies that Proposition 1. kvx kC(QT ) ≤ 2K.

Let v be the function defined by (22), then kvkC(QT ) ≤ S ∗ ,

Since u is a classical solution of (17)-(18), v is a classical solution of the following problem.   vt = d(x, t, v, vx )vxx + f (x, t, v, vx ), x ∈ (−h, h), t > 0, v(±h, t) = 0, t > 0, (23)  v(x, 0) = v0 (x) := u0 (x) − ϕ(x, a∗ ), x ∈ [−h, h],

where

a(x, y + ct + ϕ) , (24) 1 + (p + ϕx )2 p a(x, y + ct + ϕ)ϕxx f (x, t, y, p) = + b 1 + (p + ϕx )2 − c. (25) 2 1 + (p + ϕx ) Therefore, when |p| ≤ 2K we have p a∗ ≤ d(x, t, y, p) ≤ a∗ , |f (x, t, y, p)| ≤ F := a∗ K + b 1 + 9K 2 + c. (26) 2 1 + 9K d(x, t, y, p) =

Lemma 3.2. There exists a positive constant ν ∈ (0, 1) independent of v, h and T such that, for any δ ∈ (0, T ), kvkC ν,ν/2([−h,h]×[δ,T ]) ≤ Cδ ,

kvx kC ν,ν/2([−h,h]×[δ,T ]) ≤ Cδ ,

where Cδ is a constant independent of v, h and T .

(27)


239

Proof. The proof is similar to that of Lemma 3.15 in [16], where even reflection and periodic extension was done, but here we carry out odd reflection and periodic extension. More precisely, we extend v to a function vˆ defined on the whole line R by ( v(x, t), −h ≤ x ≤ h, vˆ(x, t) := −v(2h − x, t), h < x ≤ 3h, vˆ(x + 4h, t) = vˆ(x, t),

(x, t) ∈ R × [0, T ].

Then vˆ solves ˆ t, vˆ, vˆx )ˆ vˆt = d(x, vxx + fˆ(x, t, vˆ, vˆx ), where ˆ t, y, p) := d(x, fˆ(x, t, y, p) :=

(

(

(x, t) ∈ R × (0, T ),

(28)

d(x, t, y, p), −h ≤ x ≤ h, d(2h − x, t, −y, p), h < x ≤ 3h,

f (x, t, y, p), −h ≤ x ≤ h, −f (2h − x, t, −y, p), h < x ≤ 3h,

and they are 4h-periodic in x ∈ R. Note that, by the boundary conditions v(±h, t) = 0, the functions vˆx , vˆxx are continuous in R × (0, T ). Moreover, w ˆ := vˆx is a weak solution of w ˆt = (P (x, t, w, ˆ w ˆx ))x , ˆ t, vˆ(x, t), p)q + fˆ(x, t, vˆ(x, t), p). When |p| ≤ 2K we have where P (x, t, p, q) := d(x, by (26), a∗ |P (x, t, p, q)| ≤ a∗ |q| + F, P (x, t, p, q)q ≥ |q|2 − K1 1 + 9K 2 for some positive constant K1 independent of h and T . Hence, applying the interior Hölder estimates for quasilinear parabolic equations of divergence form (see [22, Theorem 2.2]) to the above equation for w, ˆ we have kˆ vx kC ν1 ,ν1 /2 (R×[δ,T ]) ≤ Cδ for any δ ∈ (0, T ), where ν1 ∈ (0, 1) does not depend on δ, v, h and T , Cδ does not depend on v, h and T . Next we derive the Hölder estimates for vˆ. Denote d1 (x, t, y, p) = a(x, y + ct + ϕ) arctan(p + ϕx ), p f1 (x, t, y, p) = b 1 + (p + ϕx )2 − c − ax (x, y + ct + ϕ) arctan(p + ϕx ) − ay (x, y + ct + ϕ)(p + ϕx ) arctan(p + ϕx ),

and extend v, d1 , f1 to vˆ, dˆ1 , fˆ1 as above for all x ∈ R, then vˆ satisfies vˆt = [dˆ1 (x, t, vˆ, vˆx )]x + fˆ1 (x, t, vˆ, vˆx ).

For (x, t, y, p) ∈ R × [0, T ] × R × [−2K, 2K], we have |dˆ1 (x, t, y, p)| ≤ a∗ |p| + a∗ K,

dˆ1 (x, t, y, p)p = a(x, y + ct + ϕ)[arctan(p + ϕx )][(p + ϕx ) − ϕx ] ≥ K2 p2 − K3 ,

|fˆ1 (x, t, y, p)| ≤ K4 (1 + |p|),

240

BENDONG LOU

where K2 , K3 , K4 are independent of v, h and T . Again, applying the interior Hölder estimates for quasilinear parabolic equations ([22, Theorem 2.2]), we see that kˆ v kC ν2 ,ν2 /2 (R×[δ,T ]) ≤ Cδ

for any δ ∈ (0, T ), where ν2 ∈ (0, 1) does not depend on δ, v, h and T , Cδ depends on δ but does not depend on v, h and T . Letting ν = min{ν1 , ν2 } ∈ (0, 1), we obtain (27).

Lemma 3.3. If v ∈ C 2+ν,1+ν/2 ([−h, h] × [0, T ]) is the solution of (23), where ν is that in (27). Then for any δ ∈ (0, T ) we have kvkC 2+ν,1+ν/2([−h,h]×[δ,T ]) ≤ Cδ ,

(29)

where Cδ is a constant independent of v, h and T .

The proof follows from interior estimates, and is similar to that of Lemma 3.17 in [16]. 3.1.3. Time-global solution. Using the previous estimates and standard theory of parabolic equations we have (see also Subsection 3.6 in [16]) Lemma 3.4. Let u0 be given as above. Then (23) has a time-global solution v(x, t) ∈ C 2+ν,1+ν/2 ([−h, h] × [0, ∞)), where ν ∈ (0, 1) is that in Lemma 3.2. Moreover, for any T > δ > 0, (29) holds. The corresponding results for u is the following. Lemma 3.5. Let u0 be given as above. Then (17)-(18) has a time-global solution u(x, t) ∈ C 2+ν,1+ν/2 ([−h, h] × [0, ∞)), where ν ∈ (0, 1) is that in Lemma 3.2. Moreover, ϕ(x, a∗ ) ≤ u(x, t) − ct ≤ ϕ(x, a∗ ) + Dh

for x ∈ [−h, h], t ≥ 0.

(30)

Cδ′

For any 0 < δ < T and any 0 < r ≤ h, there exists independent of u, h, r and T , such that kukC 2+ν,1+ν/2([−r,r]×[δ,T ]) ≤ Cδ′ + rK + cT. (31)

Proof. The global existence and inequalities (30) follow from Lemma 3.4 and (i) of Lemma 3.1, respectively. Recalling u(x, t) = v(x, t) + ct + ϕ(x, a∗ ) and using (29) we have kukC 2+ν,1+ν/2([−r,r]×[δ,T ])

≤ kvkC 2+ν,1+ν/2([−r,r]×[δ,T ]) + cT + c + kϕ(·, a∗ )kC 2+ν ([−r,r]) ≤ Cδ + cT + c + rK + K5 ,

where rK is an upper bound of kϕ(·, a∗ )kC([−r,r]) and K5 denotes kϕkC 2+ν ([−r,r]) − kϕkC([−r,r]), which is independent of u, h, r and T . Therefore, (31) holds for Cδ′ = Cδ + c + K5 . 3.1.4. Existence of entire solution. We now use the time-global solution to construct an entire solution of (17). In other words, for any ξ ∈ R, we seek for a function U (x, t) defined on [−h, h] × (−∞, ∞) and it satisfies  p Uxx  Ut = a(x, U ) + b 1 + Ux2 , x ∈ (−h, h), t ∈ R, (32) 1 + Ux2  U (±h, t) = ct + ξ, t ∈ R. In what follows, we also write (32) as (3.18)ξ and sometimes write its entire solution as U (x, t; ξ).


241

Lemma 3.6. Problem (32) has a solution U (x, t; ξ) ∈ C 2+ν,1+ν/2 ([−h, h] × R) for some ν ∈ (0, 1) independent of U and h. In addition, U has the following properties: (i) ϕ(x, a∗ ) + η ≤ U (x, t) − ct ≤ ϕ(x, a∗ ) + Dh + η for all (x, t) ∈ [−h, h] × R, where η = ξ − ϕ(h, a∗ ), (ii) |Ux (x, t)| ≤ K for all (x, t) ∈ [−h, h] × R, (iii) there exists a δ0 > 0 such that Ut (x, t) > δ0 for all (x, t) ∈ [−h, h] × R, (iv) for any T > 0 and any 0 < r ≤ h, kU kC 2+ν,1+ν/2([−r,r]×[−T,T ]) ≤ C +Kr+cT , where C is a constant independent of U, h, r and T . Proof. We use renormalization method as we did in [16]. Choose u0 (x) ≡ ϕ(x, a∗ ) and consider the problem (17)-(18). By Lemma 3.5, a global classical solution u(x, t) exists. Moreover, ut (x, t) ≥ 0 by comparison principle. For any n ∈ N, denote tn = nL/c and un (x, t) := u(x, t + tn ) − nL, where L is the period of a(x, y) in y. Then un satisfies  p (un )xx  + b 1 + (un )2x , x ∈ (−h, h), t > −tn , (un )t = a(x, un ) 1 + (un )2x  un (±h, t) = ct + ϕ(h, a∗ ), t > −tn . By (30) we have

ϕ(x, a∗ ) ≤ un (x, t) − ct ≤ ϕ(x, a∗ ) + Dh .

(33)

kun kC 2+ν,1+ν/2([−r,r]×[−T,T ]) ≤ C + Kr + cT,

(34)

For any given T > 0 and any given r ∈ (0, h], when tn > T + 1, un is defined on [−h, h] × [−T − 1, T + 1], and by (31) (choose δ = 1) we have for some ν ∈ (0, 1) and C, both are independent of un , h, r and T . Hence there exist ni → ∞ and a function U1 ∈ C 2+ν,1+ν/2 ([−h, h] × [−T, T ]) such that uni (x, t) → U1 (x, t) in C 2,1 ([−h, h] × [−T, T ]).

Taking T → ∞ and using Cantor’s diagonal argument, we have U1 (x, t) ∈ C 2+ν,1+ν/2 ([−h, h] × R) and a subsequence {nij } of {ni } such that, as j → ∞, unij (x, t) → U1 (x, t)

in C 2,1 ([−h, h] × [−T, T ]) for any T > 0.

Thus U1 satisfies the equation in (3.18)ξ and boundary conditions U1 (±h, t) = ct + ϕ(h, a∗ ). Define U (x, t; ξ) = U1 (x, t + τ ) with τ = [ξ − ϕ(h, a∗ )]/c. Then U is an entire solution of (3.18)ξ and it satisfies (i). The conclusions in (ii) and (iv) follow from |(un )x | ≤ K and (34), respectively. (iii). It is easily seen that Ut (x, t) ≥ 0 for all (x, t) ∈ [−h, h]×R since ut (x, t) ≥ 0. Suppose there exist sequences {xn }n∈N and {τn }n∈N such that Ut (xn , τn ; ξ) → 0 as n → ∞. We may assume xn → x∞ for some x∞ ∈ [−h, h]. For each n ∈ N, let kn ∈ Z be an integer satisfying kn L ≤ U (±h, τn ; ξ) = cτn < (kn + 1)L, and let Un (x, t) := U (x, t + τn ; ξ) − kn L. Then Un is an entire solution of (3.18)cτn+ξ−kn L . Since cτn + ξ − kn L ∈ [ξ, ξ + L), we have −Kh + ξ ≤ Un (x, t) − ct = Un (x, t) − Un (±h, t) + (cτn + ξ − kn L) ≤ Kh + ξ + L.

By a similar argument as above, there exist a subsequence {nj } of {n}, ξ ∗ ∈ [ξ, ξ+L] e ∈ C 2+ν,1+ν/2 ([−h, h]×R) such that cτnj +ξ−knj L → ξ ∗ , {Unj }j∈N converges and U

242

BENDONG LOU

e in the topology of C 2,1 ([−h, h]×[−T, T ]) for any T > 0. Moreover, U et (x, t) ≥ 0 to U for all (x, t) ∈ [−h, h] × R, and since et (x∞ , 0) = lim ∂Unj (xnj , 0; ξ) = lim ∂U (xnj , τnj ; ξ) = 0. U j→∞ ∂t j→∞ ∂t

et (±h, t) ≡ c 6= 0. On the other hand, x∞ ∈ It is easily seen that x∞ 6= ±h since U et ≡ 0, which also contradicts the (−h, h) and the strong maximum principle yields U e facts Ut (±h, t) ≡ c.

3.1.5. Uniqueness of entire solution. Now we prove that U (x, t; ξ1 ) is a time shift of U (x, t; ξ2 ). For this purpose, it suffices to show that they are both time shift of U (x, t; 0). In this subsection, we use U (x, t) again to denote U (x, t; 0) and assume that W (x, t) ≡ U (x, t; ξ1 ) is an entire solution of (3.18)ξ1 for some ξ1 . Assume also that W satisfies the properties (i)-(iv) in Lemma 3.6. Define ( ) ∃ a ∈ R such that, for x ∈ [−h, h], ΛU,W (t) := inf Λ > 0 . U (x, t + a) ≤ W (x, t) ≤ U (x, t + a + Λ)

It is not difficult to see by (iii) of Lemma 3.6 that the infimum is attained for some a and Λ. In fact, for each fixed t ∈ R, there exists a biggest a = a(t) such that U (x, t + a(t)) ≤ W (x, t), and the equality holds at some x. On the other hand, for this a(t), there exists a smallest Λ = Λ(t) such that W (x, t) ≤ U (x, t + a(t) + Λ(t)), and the equality holds at some x. Therefore, ΛU,W (t) is nothing but this Λ(t). Lemma 3.7. The function ΛU,W (t) has the following properties: (i) ΛU,W (t) is monotone decreasing and is bounded in t ∈ R. (ii) If ΛU,W (t0 ) = 0 for some t0 , then there exists a ∈ R such that U (·, t + a) ≡ W (·, t) for t ≥ t0 . If ΛU,W (t0 ) > 0 for some t0 , then ΛU,W (t) is positive and is strictly decreasing in t < t0 .

Proof. (i) By the definition of ΛU,W , for each fixed t ∈ R, there exists a(t) ∈ R such that U (x, t + a(t)) ≤ W (x, t) ≤ U (x, t + a(t) + ΛU,W (t)),

x ∈ [−h, h].

(35)

Therefore, it follows from the comparison principle that for any s > 0, U (x, t + s + a(t)) ≤ W (x, t + s) ≤ U (x, t + s + a(t) + ΛU,W (t)),

x ∈ [−h, h].

This implies ΛU,W (t + s) ≤ ΛU,W (t) for s > 0. Next, by (ii) of Lemma 3.6 we have max U (x, t) − min U (x, t) ≤ 2Kh,

|x|≤h

|x|≤h

max W (x, t) − min W (x, t) ≤ 2Kh.

|x|≤h

|x|≤h

(36)

By the definition of ΛU,W , there exist x1 , x2 ∈ [−h, h] and a ∈ R satisfying U (x1 , t + a) = W (x1 , t),

W (x2 , t) = U (x2 , t + a + ΛU,W (t)).

In view of this and (36), we have U (x, t + a + ΛU,W (t)) − U (x, t + a) ≤ U (x2 , t + a + ΛU,W (t)) − U (x1 , t + a) + 4Kh = W (x2 , t) − W (x1 , t) + 4Kh ≤ 6Kh.

On the other hand, Lemma 3.6 (iii) implies that U (x, t+a+ΛU,W (t))−U (x, t+a) ≥ δ0 ΛU,W (t). Hence 0 ≤ ΛU,W (t) ≤ 6M h/δ0.


243

(ii) The former statement is obvious. Suppose that ΛU,W (t0 ) > 0 for some t0 . Then by (i), ΛU,W (t) > 0 for any fixed t < t0 . Therefore (35) and the strongly comparison principle yield U (x, t + s + a(t)) < W (x, t + s) < U (x, t + s + a(t) + ΛU,W (t)),

x ∈ [−h, h]

for all s > 0. By the continuity of U (x, t) in t, there exists ε = ε(t, s) > 0 small such that U (x, t + s + a(t) + ε) < W (x, t + s) < U (x, t + s + a(t) + ΛU,W (t) − ε),

x ∈ [−h, h].

This means that at time t + s, a = a(t) + ε and Λ = ΛU,W (t) − 2ε satisfy the inequalities in the definition of ΛU,W (t + s), so ΛU,W (t + s) ≤ ΛU,W (t) − 2ε. This proves statement (ii). Lemma 3.8. W (x, t) is a time-shift of U (x, t). Proof. We only need to show that ΛU,W (t) = 0 for all t ∈ R. Suppose that ΛU,W (t0 ) > 0 for some t0 ∈ R. Then Lemma 3.7 (i) implies that ΛU,W (t) converges to some Λ > 0 as t → −∞. For each n ∈ N, let ℓn be an integer satisfying ℓn L ≤ U (±h, −n) = −cn < (ℓn + 1)L. Define Un (x, t) := U (x, t − n) − ℓn L,

Wn (x, t) := W (x, t − n) − ℓn L.

Since U is an entire solution of (3.18)0 , we see that Un is a solution of (3.18)−cn−ℓnL . Moreover, −cn − ℓn L ∈ [0, L), and |Un (x, t) − ct| ≤ |Un (x, t) − Un (±h, t)| + | − cn − ℓn L| ≤ Kh + L. So, for each n, Un satisfies the properties (i)-(iv) in Lemma 3.6 (the constant C in (iv) maybe different from the former one, but it is independent of n). Similarly, Wn is a solution of (3.18)ξ1 −cn−ℓn L , ξ1 − cn − ℓn L ∈ [ξ1 , ξ1 + L) and each Wn satisfies the properties in Lemma 3.6 (with different C in (iv) which is independent of n). A diagonal argument as in the proof of Lemma 3.6 shows that there exist a subsequence nj → ∞ (j → ∞) and two entire solutions U∞ and W∞ such that, as j → ∞, Unj → U∞ , Wnj → W∞ in C 2,1 ([−h, h] × [−T, T ])

for any T > 0. Moreover, we see that ΛUnj ,Wnj (t) → ΛU∞ ,W∞ (t) as j → ∞. On the other hand, ΛUnj ,Wnj (t) = ΛU,W (t − nj ) → Λ as j → ∞. Therefore ΛU∞ ,W∞ (t) ≡ Λ for all t ∈ R. This, however, contradicts Lemma 3.7 (ii) and the fact that Λ > 0. Thus we have ΛU,W (t) = 0 for all t ∈ R, and hence there exists a constant a such that U (x, t + a) ≡ W (x, t) for all (x, t) ∈ [−h, h] × R. 3.1.6. Existence and uniqueness of periodic traveling wave. Theorem 3.9. Let c > b > 0. Then, for any h > 0 and ξ ∈ R, (3.18)ξ has a unique “V”-like periodic traveling wave U (x, t; ξ) ∈ C 2+ν,1+ν/2 ([−h, h] × R) (ν ∈ (0, 1)). Moreover, U has average speed c and satisfies properties (i)-(iv) in Lemma 3.6. Proof. We obtained an entire solution U (x, t; ξ) in Lemma 3.6. By Lemma 3.8, it is easily seen that U (x, t; ξ) + L ≡ U (x, t; ξ + L) = U (x, t + T ; ξ) for some T , that is, U (x, t; ξ) is a periodic traveling wave. Moreover, (i) of Lemma 3.6 implies that the average speed of U is nothing but c, hence L = cT .

244

BENDONG LOU

Remark 1. We study the asymptotic shape of “V”-like periodic traveling waves as c > b > 0 and c → 0. For any fixed h > 0 and given ξ ∈ R, Theorem 3.1 gives a “V”-like periodic traveling wave U which satisfies ϕ(x, a∗ ) + η ≤ U (x, t) − ct ≤ ϕ(x, a∗ ) + Dh + η

for (x, t) ∈ [−h, h] × R,

where η = ξ − ϕ(h, a∗ ). On the other hand, Case 3 in Section 2 and (20) show that, for x ∈ [−h, h], 0 ≤ ϕ(x, a∗ ) ≤ ϕ(x, a∗ ) + Dh ≤ ϕ(h, a∗ ) + Dh = ϕ(h, a∗ ) → 0

as (b, c) → (0, 0).

So, as c > b > 0 and c → 0, U (x, t) − ct → η uniformly on [−h, h] × R. In other words, the graph of U approaches horizontal line. 3.2. “V”-like periodic traveling waves in the whole plane. We have obtained “V”-like periodic traveling waves in band domains (x, u) ∈ [−h, h] × R . In this subsection we show that there exists a “V”-like periodic traveling wave with average speed c, whose profile spans on the whole R. Proof of Main Theorem. For each m ∈ N and each ξ ∈ R, by Theorem 3.1, (3.18)ξ with h = m has a unique “V”-like periodic traveling wave U (x, t; ξ) ∈ C 2+ν,1+ν/2 ([−m, m] × R), which satisfies the properties (i)-(iv) in Lemma 3.6. Since |U (0, 0; ξ) − U (±m, 0; ξ)| = |U (0, 0; ξ) − ξ| ≤ Km, there exists a unique ξ = ξm ∈ [−Km, Km] such that U (x, t; ξm ) satisfies the normalized condition U (0, 0; ξm ) = 0. In what follows, denote Um (x, t) := U (x, t; ξm ), then Um (0, 0) = 0,

Um (±m, t) = ct + ξm .

By (i) of Lemma 3.6, ϕ(x, a∗ ) + ηm ≤ Um (x, t) − ct ≤ ϕ(x, a∗ ) + Dm + ηm

(37)

where ηm = ξm − ϕ(m, a∗ ). Taking (x, t) = (0, 0) we have ηm ≤ 0 ≤ Dm + ηm . Hence −S ∗ ≤ −Dm ≤ ηm ≤ 0. By (37) again we have, 0 ≤ Um (x, t) − ct − ϕ(x, a∗ ) − ηm ≤ ϕ(x, a∗ ) − ϕ(x, a∗ ) + Dm ≤ S ∗ .

Therefore, for any given r > 0, when m > r we have

kUm kC 2+ν,1+ν/2([−r,r]×[−r,r]) ≤ cr + Kr + |ηm | + C ≤ cr + Kr + S ∗ + C, e (x, t) ∈ where ν, C are independent of Um , m and r. Therefore, there exist U 2+ν,1+ν/2 C ([−r, r] × [−r, r]) and {mi } such that, as i → ∞, ηmi → η∗

and

e U mi → U

in C 2,1 ([−r, r] × [−r, r]).

Note that, by (20), the definitions of S ∗ and ϕ(x, a), we have Dmi → S ∗ as i → ∞. e t) ∈ Now, taking r → +∞ and using Cantor’s diagonal argument, there exist U(x, C 2+ν,1+ν/2 (R × R) and subsequence of {mi } (write it as {mi } again), such that, as e locally uniformly in C 2,1 (R2 ). Therefore U e is a solution of (5) i → ∞, Umi → U and by (37) it satisfies e (x, t) − ct ≤ ϕ(x, a∗ ) + S ∗ + η∗ ϕ(x, a∗ ) + η∗ ≤ U

(38)

b t) = U e (x, t − τ∗ ) with τ∗ = η∗ /c, then U b satisfies (8). Define U(x, e b Next we show that U (and so U ) is a periodic traveling wave. Since Um satisfies (32) with h = m and Um (x, t) + L = Um (x, t + L/c) for x ∈ [−m, m], we have e (x, t) + L = lim Umi (x, t) + L = lim Umi (x, t + L/c) = U(x, e t + L/c). U i→∞

i→∞


245

b is a periodic traveling wave whose graph is a “V”-like curve for Consequently, U each t ∈ R . This proves the Main Theorem. 4. Other periodic traveling waves in band domains. 4.1. Cup-like periodic traveling waves. From Section 2 we know that, in Cases 2, 7, 8, 9 and 10, (9) with a = a∗ has cup-shaped traveling wave ϕ(x, a∗ ) + ct on x ∈ (−xa∗ , xa∗ ) with ϕ(0, a∗ ) = 0, where Z π a∗ π a∗ b 2 ds + . xa∗ = 2c c 0 c cos s − b Similarly, (9) with a = a∗ has cup-shaped traveling wave ϕ(x, a∗ ) + ct on x ∈ (−xa∗ , xa∗ ), where xa∗ is defined as xa∗ (replacing a∗ by a∗ ), ϕ(0, a∗ ) = 0. Moreover, xa∗ < xa∗ . For any given h < xa∗ , taking Dh > 0 such that ϕ(±h, a∗ ) = ϕ(±h, a∗ ) + Dh holds we consider the problem (17). Similar argument as in Subsection 3.1 gives the following Theorem 4.1. If the pair (b, c) belongs to one of Cases 2, 7, 8, 9 and 10, and if h < xa∗ , then problem (17) (Dirichlet boundary conditions) has a unique cup-like periodic traveling wave U (x, t) ∈ C 2+ν,1+ν/2 ([−h, h] × R) (ν ∈ (0, 1)), which has average speed c and satisfies (i), (ii) and (iv) of Lemma 3.6 with K being replaced by ϕx (h, a∗ ), it also satisfies (iii) of Lemma 3.6 when c > 0. Remark 2. We study the asymptotic shape of cup-like periodic traveling waves as (b, c) → (0, 0). We only consider Cases 8, 9, 10 since Cases 2 and 7 are discussed similarly. From Section 2 we know that Z π c − b aπ ab 2 a ds . xa = + , ϕ(±xa , a) = log 2c c 0 c cos s − b c b

Denote xa = xa (b, c) and consider (10) for (b, c) being replaced by ( λb , λc ) (λ > 1): 1 + ψ2 c bp ψx = − 1 + ψ2 , (39) a λ λ which, when we write x = λz, is equivalent to p 1 + ψ2 c − b 1 + ψ2 . ψz = a

(40)

Clearly, the maximum existence interval of (40) is (−xa (b, c), xa (b, c)), and so the maximum existence interval of (39) is (−xa ( λb , λc ), xa ( λb , λc )) with xa ( λb , λc ) = λxa (b, c), which tends to +∞ as λ → +∞. This implies that, when c > 0, b < 0 and (b, c) → (0, 0) we have xa (b, c) → +∞. Also, ϕ(±xa , a) → +∞. For any given a, ψ is increasing in c and −b, and for any fixed h < xa , 0 ≤ ϕ(x, a) ≤ ϕ(h, a) → 0 as (b, c) → (0, 0). Applying this result to the inequalities (i) of Lemma 3.6 we have, as (b, c) → (0, 0), U (x, t) − ct → η uniformly on [−h, h] × R. Finally, while (b, c) → (0, 0), xa∗ → +∞, so h can be chosen as large as possible. Therefore, the graphs of the periodic traveling waves U (x, t)−ct approach horizontal lines, uniformly on [−h, h] × R for any h > 0.

246

BENDONG LOU

4.2. Cap-like periodic traveling waves. In Case 5 and Case 6, (9) with a = a∗ has cap-shaped traveling wave ϕ(x, a∗ ) + ct on x ∈ (−xa∗ , xa∗ ) with ϕ(0, a∗ ) = 0, where Z π a∗ π ds a∗ b 2 xa∗ = − + , 2c c 0 b − c cos s Similarly, (9) with a = a∗ has cap-shaped traveling wave ϕ(x, a∗ ) + ct on x ∈ (−xa∗ , xa∗ ) with ϕ(0, a∗ ) = 0, where xa∗ is defined as xa∗ (replacing a∗ by a∗ ). Moreover, xa∗ < xa∗ . For any given h < xa∗ , taking Dh > 0 such that ϕ(±h, a∗ ) = ϕ(±h, a∗ ) − Dh we consider the problem (17). Similar argument as above gives the following Theorem 4.2. If the pair (b, c) belongs to Case 5 or Case 6, and if h < xa∗ , then (17) (Dirichlet boundary conditions) has a unique cap-like periodic traveling wave U (x, t) ∈ C 2+ν,1+ν/2 ([−h, h] × R) (ν ∈ (0, 1)), which has average speed c and satisfies ϕ(x, a∗ ) − Dh + η ≤ U (x, t) − ct ≤ ϕ(x, a∗ ) + η,

(x, t) ∈ [−h, h] × R,

(41)

for some η ∈ R. It also satisfies (ii) and (iv) in Lemma 3.6 with K being replaced by |ϕx (h, a∗ )|, satisfies (iii) in Lemma 3.6 when c > 0. Remark 3. Similar conclusion as in Remarks 1 and 2 hold. In other words, for b > c ≥ 0, while (b, c) → (0, 0), xa∗ → +∞, and so h can be chosen as large as possible. The graphs of the cap-like periodic traveling waves U (x, t) approach horizontal lines as (b, c) → (0, 0), uniformly on [−h, h] × R for any h > 0. Remark 4. As we have seen in Sections 3 and 4, the Dirichlet boundary condition on finite intervals is helpful not only for the a priori estimates in the existence stage, but also for the description of the shape of the profile (see (i) of Lemma 3.6, (8) and (41)). We remark that Neumann boundary conditions on finite intervals have clear physical meanings, they are also good choice if one only consider the existence of periodic traveling waves in band domains. The existence results in Theorems 3.1, 4.1 and 4.2 remain hold if the Dirichlet boundary conditions are replaced by Neumann boundary conditions like (15), the proof are also similar. However, the shape of the profile seems not clear if Neumann boundary condition are imposed. One reason is that, under Neumann boundary conditions, we have no prescribed speed c. So even the upper and lower solutions can be constructed, they have different traveling speeds, and they are not sharp enough to give sufficient information on the shape of the profile. Combining with the reasons we have stated in the beginning of Section 3, we adopted Dirichlet boundary conditions in this paper. 5. Bernshte˘ın’s problem and De Giorgi’s conjecture. Recently, many authors studied the following spatially heterogeneous reaction-diffusion equation 1 Ut = ∇(A(x)∇U) + 2 B(x)U Z 2 (x) − U 2 , x ∈ Rn , t > 0, (42) ε where A, B and Z are bounded, smooth functions with positive infimums, x =(x′ , xn ) and x′ = (x1 , x2 , · · · , xn−1 ). In the special case where Z ≡ 1, the nonlinearity is a typical cubic function. Formally, when 0 < ε ≪ 1, (42) has exactly three stationary solutions: Z+ , 0 and Z− with Z± ≈ ±Z. It was shown in [1, 7, 10, 14, 17] that, sharp internal layers between Z+ and Z− appear in a very fast time scale, and then the layers (or,


247

equivalently, the level set of U: Γt := {x | U(x, t) = 0}) propagate in the following way: V = −(n − 1)a(x)κ + c(x)∇d(x) · n for x ∈ Γt , t > 0, (43)

where n is the normal direction to Γt , V and κ are as above, a, c, d are bounded, smooth functions with inf a > 0. In what follows, denote by (42)0 the spatially homogeneous version of (42) (A, B, Z are constants), denote by (43)0 the spatially homogeneous version of (43) (a, c, d are constants). The well known Bernshte˘ın’s problem is about the stationary hypersurface of (43)0 : Bernshte˘ın’s Problem: Let g : Rn−1 → R be a C 2 function, S(g) ⊂ Rn be the graph of g. If κ = 0 on S(g), then S(g) is a hyperplane.

It is known that the answer to this problem is positive for n ≤ 8 (E. De Giorgi, Almgren, Simons) and negative for n ≥ 9 (E. Bombieri -De Giorgi - E. Giusti). Clearly, Bernshte˘ın’s problem is related to the shape of the stationary solution of (42)0 , for which, De Giorgi [9] proposed the following conjecture: De Giorgi’s Conjecture: Assume that U is a stationary, entire solution (defined ∂U > 0. Then, at least for for all x ∈ Rn ) of (42)0 , that it satisfies |U| ≤ 1 and ∂x n n ≤ 8, the level sets of U must be hyperplanes. This conjecture was partially proved recently in [11], [21] and references therein. Remarks 1, 2 and 3 above indicate that we can propose some spatially heterogeneous versions for Bernshte˘ın’s Problem and De Giorgi’s Conjecture. Spatially Heterogeneous Bernshte˘ın’s Problem: Let g : Rn−1 → R be a C 2 function, S(g) ⊂ Rn be the graph of g. Assume that S(g) is a stationary solution of (43). Then at least for n ≤ 8, there exists a linear function g0 : Rn−1 → R such that (i) g − g0 is quasi-periodic if a, c, d are quasi-periodic; (ii) g − g0 is almost periodic if a, c, d are almost periodic (in the sense of Bohr, for example). In almost periodic case (ii), a primary analysis indicates that some additional assumptions maybe needed: for any ball B ⊂ Rn , Z Z Z [a(x) − a0 ]dx, [c(x) − c0 ]dx, [d(x) − d0 ]dx (44) B

B

B

are bounded by M (independent of B), where a0 , c0 and d0 are the arithmetic means of a, c and d, respectively. Spatially Heterogeneous De Giorgi’s Conjecture. Assume that U is an entire, stationary solution of (42), that it satisfies Z− (x) ≤ U(x) ≤ Z+ (x),

lim

xn →±∞

U(x) = Z± (x).

(45)

Then at least for n ≤ 8, the 0-level set of U: Γ := {x | U(x) = 0 } must be the graph of a function xn = g(x′ ). Moreover, there exists a linear function xn = g0 (x′ ) such that (i) g − g0 is quasi-periodic if A, B, Z are quasi-periodic; (ii) g − g0 is almost periodic if A, B, Z are almost periodic and if they satisfy similar conditions as in (44).

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Some further conditions maybe also needed. For example, in the periodic case, ∂U denote by X the common period of A, B and Z in xn -direction. One sees that ∂x > n 0 as in De Giorgi’s conjecture is not necessarily to be true, instead, U(x′ , xn ) < U(x′ , xn + X) (x ∈ Rn ) may play a similar role and be used as an additional condition for the conjecture. We now discuss the spatially heterogeneous FitzHugh-Nagumo equations Ut = ∇(A(x)∇U) + ε−2 B(x)U(Z 2 (x) − U 2 ) − D(x)V, x ∈ Rn , t > 0, Vt = ∇(α(x)∇V) + β(x)U − γ(x)V, x ∈ Rn , t > 0. (46) It is known that, when 0 < ε ≪ 1, the solution U of (46) behaviors like that of (42). In fact, from (46) one also derives the law of the motion of the level set of U, which is similar to (43) (cf. [1, 14, 17]). For such reasons, we propose an analogue of De Giorgi’s conjecture for (46). A Conjecture for FitzHugh-Nagumo Equations. Let A, B, D, Z, α, β, γ be positive, almost periodic functions (in the sense of Bohr). Assume that 0 < ε ≪ 1 and that (U, V) is an entire stationary solution of (46). Assume also that Z− (x) ≤ U(x) ≤ Z+ (x),

lim

xn →±∞

U(x) = Z± (x).

(47)

Then at least for n ≤ 8, the 0-level set of U: Γ := {x | U(x) = 0 } must be the graph of a function xn = g(x′ ). Moreover, there exists a linear function xn = g0 (x′ ) such that g − g0 is a constant (resp. quasi-periodic function, almost periodic function) when A, B, D, Z, α, β, γ are constants (resp. quasi-periodic functions, almost periodic functions satisfying similar conditions as in (44)). In case n = 2, hypersurface Γt reduces to a curve in the plane. Spatially heterogeneous Bernshte˘ın’s Problem claim that any entire stationary curve of (43) must be quasi-periodic (or, almost periodic) oscillation of a straight line. Remarks 1, 2 and 3 above give indication for the possibility of this conjecture. Acknowledgments. The author is grateful to the referees for helpful suggestions. REFERENCES [1] M. Alfaro, “Convection-Reaction-Diffusion Systems and Interface Dynamics,” Ph.D thesis, University of Paris XI, 2006. [2] S. J. Altschuler and L. F. Wu, Convergence to translating solitons for a class of quasilinear parabolic equations with fixed angle of contact to a boundary, Math. Ann., 295 (1993), 761– 765. [3] S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101–111. [4] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II - biological invasions and pulsating traveling fronts, J. Math. Pure Appl. (9), 84 (2005), 1101–1146. [5] P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Physica D, 94 (1996), 205–220. [6] P. K. Brazhnik and V. A. Davydov, Non-spiral autowave structures in unrestricted excitable media, Phys. Lett. A, 199 (1995), 40–44. [7] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116–141. [8] K. S. Chou and X. P. Zhu, “The Curve Shortening Problem,” Chapman & Hall/CRC, New York, 2001.


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Received for Publication February 2007. E-mail address: [email protected]