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

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, 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.
Keywords: ray methods, asymptotic analysis., wave propagation, Reaction-diffusion.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35C2.
Citation: John R. King. Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity. Networks & 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. doi: 10.1093/imamat/3.3.266. Google Scholar

[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. doi: 10.1093/qjmam/hbq017. Google Scholar

[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. doi: 10.1016/S0167-2789(00)00068-3. Google Scholar

[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. doi: 10.1512/iumj.1989.38.38007. Google Scholar

[5]

J. Smoller, "Linear Elastic Waves,", Cambridge University Press, (2001). doi: 10.1017/CBO9780511755415. Google Scholar

[6]

M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, Ann. Prob., 13 (1985), 639. doi: 10.1214/aop/1176992901. Google Scholar

[7]

John King, "Mathematical Aspects of Semiconductor Process Modelling,", DPhil Thesis, (1986). Google Scholar

[8]

J. R. King, High concentration arsenic diffusion in crystalline silicon: An asymptotic analysis,, IMA J. Appl. Math., 38 (1987), 87. doi: 10.1093/imamat/38.2.87. Google Scholar

[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). doi: 10.1103/PhysRevE.68.041105. Google Scholar

[10]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,", American Mathematical Society, (1994). Google Scholar

[11]

J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296. Google Scholar

show all references

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. doi: 10.1093/imamat/3.3.266. Google Scholar

[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. doi: 10.1093/qjmam/hbq017. Google Scholar

[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. doi: 10.1016/S0167-2789(00)00068-3. Google Scholar

[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. doi: 10.1512/iumj.1989.38.38007. Google Scholar

[5]

J. Smoller, "Linear Elastic Waves,", Cambridge University Press, (2001). doi: 10.1017/CBO9780511755415. Google Scholar

[6]

M. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, Ann. Prob., 13 (1985), 639. doi: 10.1214/aop/1176992901. Google Scholar

[7]

John King, "Mathematical Aspects of Semiconductor Process Modelling,", DPhil Thesis, (1986). Google Scholar

[8]

J. R. King, High concentration arsenic diffusion in crystalline silicon: An asymptotic analysis,, IMA J. Appl. Math., 38 (1987), 87. doi: 10.1093/imamat/38.2.87. Google Scholar

[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). doi: 10.1103/PhysRevE.68.041105. Google Scholar

[10]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,", American Mathematical Society, (1994). Google Scholar

[11]

J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296. Google Scholar

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