Influence of the Domain Boundary on the Speeds of Traveling Waves

Let H > 0 be a constant, g ≥ 0 be a periodic function and ( ) ( ) { }  ∈ H Ω = , < + , x y x g y y . We consider a curvature flow equation V = κ + A in Ω, where for a simple curve ⊂ Ω t γ , V denotes its normal velocity, κ denotes its curvature and A > 0 is a constant. [1] proved that this equation has a periodic traveling wave U, and that the average speed c of U is increasing in A and H, decreasing in max g' when the scale of g is sufficiently small. In this paper we study the dependence of c on A, H, max g' and on the period of g when the scale of g is large. We show that similar results as [1] hold in certain weak sense.


Introduction
We study traveling waves for a curvature-driven motion of plane curves in a band domain Ω.The law of motion of the curve is given by on , where t γ ⊂ Ω is a simple, smooth curve, V denotes its normal velocity, κ denotes its curvature and A is a positive constant representing a driving force.The band domain Ω is defined as the following.Set , 0 0, is 1-periodic, max 1 .
y g y g y C g y g g g y For some g ∈ we define ( )  ∂ Ω (resp.+ ∂ Ω ).By a solution of (1) we mean a time-dependent simple, smooth curve t γ in Ω which satisfies (1) and con- tacts ± ∂ Ω perpendicularly.Equation (1) appears as a certain singular limit of an Allen-Cahn type nonlinear diffusion equation under the Neumann boundary conditions.The curve t γ represents the interface between two different phases (see, e.g., [1]- [4] for details).In physics, chemistry and many other fields, an interface may propagate in a domain with obstacles, say, with obstacles lying in several lines.The motion of the interface between two adjacent lines is then like the propagation of t γ in Ω in our problem.Hence the undulation of the boundary of Ω can be regarded as effect of obstacles and so it can be in any size.[1] studied the homogenization limit of this problem (as 0 p → ), we will consider the case where p is large.
To avoid sign confusion, the normal to the curve t γ will always be chosen toward the upper region, and the sign of the normal velocity V and the curvature κ will be understood in accordance with this choice of the normal direction.Consequently, κ is negative at those points where the curve is concave while it is positive where the curve is convex (see Figure 1).
In the case where t γ is expressed as a graph of a function with the boundary conditions < in the definition  prevent t γ from developing singularities near the boundary ± ∂ Ω (cf.[1]).Denote

( )
tan : max and call g α the maximum opening angle of ( ) , , H g p Ω , or, of g.Then π 0, 4 , ; , , , , , µ ∈ (also write as ( ) , U x t for simplicity) of (3)-( 4) is called a periodic traveling wave if it satisfies ( ) ( )  , , , ,  : .In [1] the authors proved that, under the condition sin g AH α > , the problem (3)-( 4) has a periodic traveling wave ( ) , U x t , it is unique under the normalization condition ( ) for all t ∈  and x where U is defined.In addition, [1] studied the homogenization limit of the average speed c.
Theorem A gives the dependence of c on A, H and g α near the homogenization limit (as 0 p → ).It is known that in the study of spatially heterogeneous problems, homogenization is a powerful method when the spatial heterogeneity is fine (for example, 0 p → in our problem) (cf.[5] [6]).On the contrary, the mathematical analysis is completely different and very difficult when the spatial heterogeneity is coarse (for example, 0 p > is large in our problem).How does the traveling wave U and its average speed c in our problem depend on the parameters , , , g A H g α and p when p is large?This is an interesting problem in physics and is also a challenging one in mathematics.Some mathematicians from Japan and France have been working on it for several years, but yet very little is known so far.Our main purpose in this paper is to study this problem by analytic and numerical methods, and try to give some answers.This paper is arranged as the following.In section 2 we list some notations and present our main theorem.In section 3 we prove the main theorem.In subsection 3.1 we prove that ( ) α .Finally, in section 4 we present some numerical simulation results, including the dependence of c on the period p of p g .

Notations and Main Results
We list some notations for convenience.For any g ∈ , 0 Clearly, N depends on , , , g A g p α and K 1 , K 2 depend on , , , g A H g α .Finally, for any π 0, 4 Here is an example, let ( ) ( ) ( ) , then we have The following is our main result.Main Theorem.Assume g ∈ and sin g AH α > . Then , where N is given by (11), ( ) ( ) ( ) ( )  .We remark that 3) of the theorem mainly states the dependence of c on α but not on g itself.In fact, for k α defined by ( 14), the conclusion of 3) holds for any k g provided max tan (with restrictions ( ) ), the exact shape of k g does not matter.
By the main theorem, ( ) , , , , g c A H g p α is increasing in continuously varying A, but it is increasing in H and decreasing in g α only in weak sense, that is, the monotonicity holds only for certain sequences.It turns out that the monotonicity for continuously varying H and g α is very difficult.In fact, we believe that 0 g c α ∂ ∂ < is not true when p is large.This is quite different from the case where 0 p → .

Proof of the Main Theorem
In this section, for any two solutions 1 u and 2 u of ( 3)-( 4), when we write 1 2 u u ≤ or 1 2 u u < we mean that the inequality holds on the common domain where 1 u and 2 u are defined.

Proof of Main Theorem 1
Assume that for some 0 x .This is possible since , 0 0, , 0 0, , 0 0, are both bounded functions.We show that First by the maximum principle (see, for example, Theorem 2 in Chapter 3 in [7]) we have This implies that the graph of , , from below except on their end points.On the other hand, if the latter happens on the right boundary, that is, there exists 1 0 t > such that ( ) , 0 and so ( ) ( ) since, otherwise we have x by Taylor's formula.But this contradicts (16).

2
A A > and using the equations of 1 U and 2 U we have ( ) ( ) we see that at the point This means that, in a small time-interval around 1 t t = , the graph of U 1 moves along the boundary + ∂ Ω faster than the graph of U 2 .This, however, contradicts the fact 1 2

U U
> and the assumption that ( ) ( ) . This proves (15).Now taking 0 x x = and 2 t T = in (15) and using the definitions of i T we have , , , , , , , , > by the definition of c in (6).This proves 1) of the Main Theorem.

Dependence of c on H
In this subsection we study the dependence of c on H and prove Main Theorem 2).Since only H is varying, for simplicity, in this part we only indicate H but omit all the other parameters in the notations Ω, U, c, B, J, K i , ••• For any ( )

U H t H U h t H k H h U H t H U h t H p H h p U h H h t H H h p U x t H H h p J H h p pJB J H h H h
and so ( )

( ) (
) . This proves 2) of the main theorem. Remark If we take 0 A = in the original problem, then to guarantee the existence of periodic traveling waves we should modify the boundary conditions a little.For example, one can consider a problem such that the curve t γ always has a positive/negative slope on the right/left boundary.In this case, a similar discussion as in [1] shows that the problem has a unique periodic traveling wave U with average speed c.Moreover 0 t U > , or equivalently, 0 xx U > .Using these results and using a similar discussion as above we can show that 0 c H ∂ ∂ < .In other words, the monotonic dependence of c on H is completely different from the case 0 A > .

Dependence of c on g and α g
In this subsection we study the dependence of c on g and g α .Similar as above, we only indicate g and g α but omit all the other parameters in the notations , , U c  for simplicity.First we note that a classical traveling wave solution of (3) (with a constant speed and a constant profile) is generally written in the form ( ) ( ) 0 , u x t x c t ϕ = + .Substituting this form into (3) yields The solution ( ) where ( ) is the average speed of the periodic traveling wave ( ) , ; ,  , ; , .  Proof of Main Theorem 3. We write g and g α as 0 g and 0 α , respectively.For any 0,1, 2, , By Lemma 1 and the definitions of ( )

<
On the other hand, replacing ( ) , in Lemma 4 we have Combining the above inequalities with (25) we have ( This proves Main Theorem 3).

Some Numerical Simulation Results
In this section we present some numerical simulation figures.Figure 2 indicates that the average speed c is strictly increasing in the basic width H of the domain.The results shown in Figure 2 and Figure 3 are partially proved in the main theorem.The dependence of c on p is very difficult, and we have no analytic result so far.

Figure 1 .
Figure 1.Domain Ω (the left one has fine boundaries, the right one has coarse boundaries).
considering the normalization and the symmetry of Ω, we impose the following initial condition:

Figure 3
Figure 3 indicates that the average speed c is strictly decreasing in the maximum opening angle α .

Figure 4
Figure4indicates that the average speed c is strictly increasing in the period p of g.

Figure 2 .
Figure 2. The monotonic dependence of c on H.

Figure 3 .
Figure 3.The monotonic dependence of c on α.

Figure 4 .
Figure 4.The monotonic dependence of c on p. .
×  .If we denote the x-coordinate of the contact point by ( ) , h H

of Main Theorem 2. Set
is a classical traveling wave of (28).So we can use comparison principle for w and ( )