Influence of the Domain Boundary on the Speeds of Traveling Waves ()
Abstract
Let H > 0 be a constant, g ≥ 0 be a periodic function and Ω ={(x, y) ||x| < H + g (y), y ∈R}. We consider a curvature flow equation V = κ + A in Ω, where for a simple curve γt
Ω, 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.
Share and Cite:
Ma, L. and Tan, J. (2014) Influence of the Domain Boundary on the Speeds of Traveling Waves.
Applied Mathematics,
5, 2088-2097. doi:
10.4236/am.2014.513203.
Conflicts of Interest
The authors declare no conflicts of interest.
References
[1]
|
Matano, H., Nakamura, K.I. and Lou, B. (2006) Periodic Traveling Waves in a Two-Dimensional Cylinder with Saw-Toothed Boundary and Their Homogenization Limit. Networks and Heterogeneous Media, 1, 537-568. http://dx.doi.org/10.3934/nhm.2006.1.537
|
[2]
|
Alfaro, M., Hilhorst, D. and Matano, H. (2008) The Singular Limit of the Allen-Cahn Equation and the FitzHugh-Nagumo System. Journal of Differential Equations, 245, 505-565. http://dx.doi.org/10.1016/j.jde.2008.01.014
|
[3]
|
Lou, B. (2007) Singular Limits of Spatially Inhomogeneous Convection-Reaction-Diffusion Equation. Journal of Statistical Physics, 129, 509-516. http://dx.doi.org/10.1007/s10955-007-9400-3
|
[4]
|
Nakamura, K.I., Matano, H., Hilhorst, D. and Schatzle, R. (1999) Singular Limits of Spatially Inhomogeneous Convection-Reaction-Diffusion Equation. Journal of Statistical Physics, 95, 1165-1185. http://dx.doi.org/10.1023/A:1004518904533
|
[5]
|
Cioranescu, D. and Donato, P. (1999) An Introduction to Homogenization. Oxford University Press, Oxford.
|
[6]
|
Cioranescu, D. and Saint Jean Paulin, J. (1999) Homogenization of Reticulated Structures. Springer-Verlag, New York.
|
[7]
|
Protter, M.H. and Weinberger, H.F. (1967) Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs, 172-173.
|