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.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.