Influence of the Domain Boundary on the Speeds of Traveling Waves

Affiliation(s)

Ningbo Shentong Energy Co., Ltd., Ningbo, China.

Department of Mathematics, Tongji University, Shanghai, China.

Ningbo Shentong Energy Co., Ltd., Ningbo, China.

Department of Mathematics, Tongji University, Shanghai, China.

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 γ_{t } _{}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.

Let

Cite this paper

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.

Ma, L. and Tan, J. (2014) Influence of the Domain Boundary on the Speeds of Traveling Waves.

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.

[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.