# American Institute of Mathematical Sciences

March  2013, 8(1): 343-378. doi: 10.3934/nhm.2013.8.343

## Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity

 1 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom

Received  March 2012 Revised  February 2013 Published  April 2013

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.
Citation: John R. King. Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity. Networks and Heterogeneous Media, 2013, 8 (1) : 343-378. doi: 10.3934/nhm.2013.8.343
##### 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.

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##### 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.
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