The Decay Rates of Traveling Waves and Spectral Analysis for a Class of Nonlocal Evolution Equations

G. Zhao¹ ; S. Ruan²

¹ Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA
² Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA

Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 142-162.

Résumé

We obtain the precise decay rates of traveling waves for a class of nonlocal evolution equations arising in the theory of phase transitions and mathematical biology. We also investigate the spectrum of the operator obtained by linearizing at such a traveling wave. The detailed description of the spectrum is established.

Détail

DOI : 10.1051/mmnp/20150610

Bibliographie

[1] H. Amann SIAM Rev. 1976 620 709

[2] F. Andreu, J. M. Mazón, J. D. Rossi, J. Toledo, Nonlocal Diffusion Problems, Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 2010.

[3] N. Apreutesei, A. Ducrot, V. Volpert Math. Model. Nat. Phenom. 2008 1 27

[4] N. Apreutesei, N. Bessonov, V. Volpert, V. Vougalter Discrete Contin. Dynam. Syst. Ser. B. 2010 537 557

[5] N. Apreutesei and V. Volpert, Properness and topological degree for nonlocal reaction-diffusion operators, Abstract Appl. Anal. (2011), 1-21.

[6] N. Apreutesei, V. Volpert Intl. Electron. J. Pure Appl. Math. 2012 53 67

[7] N. Apreutesei, V. Volpert Topol. Methods Nonlinear Anal. 2014 215 229

[8] D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in “Nonlinear Diffusion”, eds. by W. E. Fitzgibbon and H. F. Walker, Research Notes in Math. 14, Pitman, London, 1977, pp. 1-23.

[9] P. W. Bates, A. Chmaj J. Stat. Phys. 1999 1119 1139

[10] P. W. Bates, F. Chen SIAM J. Math. Anal. 2006 116 126

[11] P. W. Bates, P.C. Fife, X. Ren, X. Wang Arch. Rat. Mech. Anal. 1997 105 136

[12] P. W. Bates, J. Han, G. Zhao Nonlinear Anal. 2006 2251 2278

[13] P. W. Bates, G. Zhao J. Math. Anal. Appl. 2007 428 440

[14] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[15] X. Chen Adv. Differential Equations 1997 125 160

[16] Z. Chen, B. Ermentrout, B. Mcleod Appl. Anal. 1997 235 253

[17] A. De Masi, E. Orlandi, E. Presutti, L. Triolo Proc. Roy. Soc. Edinburgh 1994 1013 1022

[18] T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, New York, 1976.

[19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.

[20] W. Huang J. Dynam. Differential Equations 2001 147 183

[21] H. J. Hupkes, S. M. Verduyn Lunel J. Dynam. Differential Equations 2005 523 572

[22] J. Mallet-Paret J. Dynam. Differential Equations 1999 1 48

[23] J. Mallet-Paret J. Dynam. Differential Equations 1999 49 127

[24] M. Miklavcic, Applied Functional Analysis and Partial Differential Equations, World Scientific, Singapore, 1998.

[25] A. Pazy Arch. Rat. Mech. Anal. 1967 105 136

[26] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in “Mathematics for Life Science and Medicine”, Vol. 2, Y. Takeuchi, K. Sato and Y. Iwasa (eds.), Springer-Verlag, New York, 2007, pp. 97-122.

[27] K. H. Schumacher J. Reine Angew. Math. 1980 54 70

[28] H. R. Thieme Discrete Contin. Dynam. Syst. 1998 73 90

[29] A. I. Volpert, Vi. A. Volpert and Vl. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translated from Russian by J. F. Heyda, Transl. Math. Monogr., Vol. 140, Amer. Math. Soc., Providence, 1994.

[30] H. F. Weinberger SIAM J. Math. Anal. 1982 353 396

[31] G. Zhao Discrete Contin. Dynam. Syst. 2009 1025 1045

[32] G. Zhao, S. Ruan J. Math. Pures Appl. 2011 627 671

[33] G. Zhao, S. Ruan J. Differential Equations 2014 1078 1147