Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity

John R. King; John R. King

doi:10.3934/nhm.2013.8.343

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 343-378. doi: 10.3934/nhm.2013.8.343

Previous Article Next Article

Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity

John R. King ^{1
,}

1.
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD

Received: 01 March 2012 Revised: 01 February 2013
Primary: 35K57; Secondary: 35C20.

We adapt (ray-based) geometrical optics approaches to encompass the formal asymptotic analysis of front propagation in a Fisher-KPP equation with slowly varying spatial inhomogeneities. The wavespeed is shown to be selected by two distinct (and fully constructive) mechanisms, depending on whether the source term is an increasing or decreasing function of the spatial variable. Canonical inner problems, analogous to those arising in the geometrical theory of diffraction, are formulated to give refined expressions for the wavefront location. Additional phenomena, notably the initiation of new fronts and the transitions that occur when the source term is a non-monotonic function of space, are shown to be amenable to the same asymptotic approaches.
- Reaction-diffusion,
- wave propagation,
- ray methods,
- asymptotic analysis
Citation: John R. King. Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity[J]. Networks and Heterogeneous Media, 2013, 8(1): 343-378. doi: 10.3934/nhm.2013.8.343

Related Papers:

Abstract

We adapt (ray-based) geometrical optics approaches to encompass the formal asymptotic analysis of front propagation in a Fisher-KPP equation with slowly varying spatial inhomogeneities. The wavespeed is shown to be selected by two distinct (and fully constructive) mechanisms, depending on whether the source term is an increasing or decreasing function of the spatial variable. Canonical inner problems, analogous to those arising in the geometrical theory of diffraction, are formulated to give refined expressions for the wavefront location. Additional phenomena, notably the initiation of new fronts and the transitions that occur when the source term is a non-monotonic function of space, are shown to be amenable to the same asymptotic approaches.

References

[1]	J. K. Cohen and R. M. Lewis, A ray method for the asymptotic solution of the diffusion equation, IMA J. Appl. Math., 3 (1967), 266-290. doi: 10.1093/imamat/3.3.266
[2]	C. M. Cuesta and J. R. King, Front propagation a heterogeneous Fisher equation: The homogeneous case is non-generic, Q. J. Mech. Appl. Math., 63 (2010), 521-571. doi: 10.1093/qjmam/hbq017
[3]	U. Ebert and W. van Saarloos, Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Physica D, 146 (2000), 1-99. doi: 10.1016/S0167-2789(00)00068-3
[4]	L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Uni. Math. J., 38 (1989), 141-172. doi: 10.1512/iumj.1989.38.38007
[5]	J. Smoller, "Linear Elastic Waves," Cambridge University Press, 2001. doi: 10.1017/CBO9780511755415
[6]	M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. Prob., 13 (1985), 639-675. doi: 10.1214/aop/1176992901
[7]	John King, "Mathematical Aspects of Semiconductor Process Modelling," DPhil Thesis, University of Oxford. 1986.
[8]	J. R. King, High concentration arsenic diffusion in crystalline silicon: An asymptotic analysis, IMA J. Appl. Math., 38 (1987), 87-95. doi: 10.1093/imamat/38.2.87
[9]	V. Méndez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media, Phys. Rev. E, 68 (2003), 041105. doi: 10.1103/PhysRevE.68.041105
[10]	A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," American Mathematical Society, 1994.
[11]	J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296

Reader Comments

Your name:*

Email:*
© 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)