Asymptotic analysis of a monostable equation in periodic media

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Matthieu Alfaro
Thomas Giletti

Abstract

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} %[4]of the well-known spreading properties \cite{Wein02}, %[32], \cite{Ber-Ham-02}, %[9],and the solution of a Hamilton-Jacobi equation.

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How to Cite
Alfaro, M., & Giletti, T. (2016). Asymptotic analysis of a monostable equation in periodic media. Tamkang Journal of Mathematics, 47(1). https://doi.org/10.5556/j.tkjm.47.2016.1872
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Author Biographies

Matthieu Alfaro

I3M, Université deMontpellier 2, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France.

Thomas Giletti

IECL, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France.

References

M.Alfaro and A. Ducrot, Sharp interface limit of the Fisher-KPP equation, Comm. Pure Appl. Anal. 11 (2012),1--18.

M.Alfaro and A. Ducrot, Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay, Discrete Contin. Dyn. Syst. Ser. B., 16(2011), 15--29.

M.Alfaro and A. Ducrot, Propagating interface in a Fisher-KPP equation with delay, Differential Integral Equations, 27(2014), 81--104.

M.Alfaro and T. Giletti, Varying the direction of propagation in monostable reaction-diffusion equations in periodic media, submitted.

M. Bardi and L. C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., (1984), 1373--1381.

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J.,61(1990),835--858.

G. Barles and P. E. Souganidis, A remark on the asymptotic behavior of the solution of the KPP equation, C. R. Acad. Sci. Paris Serie I, 319(1994), 679--684.

G. Barles and P.E.Souganidis, A new approach to front propagation problems : theory and applications, Arch. Rat. Mech. Anal., 141(1998), 237--296.

H.Berestycki and F. Hamel,Front propagation in periodic excitable media}, Comm. Pure Appl. Math., 55(2002), 949--1032.

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51(2005), 5--113.

H. Berestycki, F. Hamel and L. Roques, it Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts, J. Math.

Pures Appl., 84(2005), 1101--1146.

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom., 33(1991), 749--786.

L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., 38(1989),141--172.

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355--369.

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab., 13(1985), 639--675.

M. I. Freidlin, Coupled reaction-diffusion equations, Ann. Probab.,19(1991), 29--57.

Y. Giga, Surface evolution equations, Monographs in Mathematics 99, Birkhauser Verlag, Basel, Boston, Berlin, 2006.

Y. Giga, Personal communication, The University of Tokyo, January 2015.

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited}, Nature, 287(1980), 17--21.

F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl., 89(2008), 355--399.

F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc., 13(2011), 345--390.

J.-B. Hiriart-Urruty and C. Lemarechal, Convex analysis and minimization algorithms.I, Grundlehren der Mathematischen Wissenschaften [Fundamental

Principles of Mathematical Sciences], 305. Springer-Verlag, Berlin,1993.

E. Hopf, Generalized solutions of non-linear equations of first order, J. Math. Mech., 14(1965), 951-973.

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de lequation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bulletin Universite d'Etat Moscou, Bjul. Moskowskogo Gos. Univ., 1937,1--26.

P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished, 1986.

P. L. Lions and P. E. Souganidis, Homogenization of viscous'' Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations, 30(2005), 335--375.

A. J. Majda and P. E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7(1994), 1--30.

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9), 92(2009), 232--262.

A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool., 2(1954), 9--65.

N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practice, Oxford University Press, 1997.

P. E. Souganidis, Front propagation: theory and applications. Viscosity solutions and applications, (Montecatini Terme, 1995), 186--242, Lecture Notes in Math., 1660, Springer, Berlin, 1997.

H.Weinberger, On spreading speed and travelling waves for growth and migration, J. Math. Biol., 45(2002), 511--548.

J. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal., 121(1992), 205--233.