×

Properness and topological degree for nonlocal reaction-diffusion operators. (English) Zbl 1215.35089

Summary: The paper is devoted to integro-differential operators, which correspond to nonlocal reaction-diffusion equations considered on the whole axis. Their Fredholm property and properness are proved. This allows one to define the topological degree.

MSC:

35K57 Reaction-diffusion equations
35R09 Integro-partial differential equations
47H11 Degree theory for nonlinear operators
47A53 (Semi-) Fredholm operators; index theories
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] S. Genieys, V. Volpert, and P. Auger, “Pattern and waves for a model in population dynamics with nonlocal consumption of resources,” Mathematical Modelling of Natural Phenomena, vol. 1, no. 1, pp. 65-82, 2006. · Zbl 1201.92055 · doi:10.1051/mmnp:2006004
[2] V. Volpert and S. Petrovskii, “Reaction-diffusion waves in biology,” Physics of Life Reviews, vol. 6, pp. 267-310, 2009.
[3] S. Ai, “Traveling wave fronts for generalized Fisher equations with spatio-temporal delays,” Journal of Differential Equations, vol. 232, no. 1, pp. 104-133, 2007. · Zbl 1113.34024 · doi:10.1016/j.jde.2006.08.015
[4] N. Apreutesei, N. Bessonov, V. Volpert, and V. Vougalter, “Spatial structures and generalized travelling waves for an integro-differential equation,” Discrete and Continuous Dynamical Systems. Series B, vol. 13, no. 3, pp. 537-557, 2010. · Zbl 1191.35041 · doi:10.3934/dcdsb.2010.13.537
[5] N. Apreutesei, A. Ducrot, and V. Volpert, “Competition of species with intra-specific competition,” Mathematical Modelling of Natural Phenomena, vol. 3, no. 4, pp. 1-27, 2008. · Zbl 1337.35068 · doi:10.1051/mmnp:2008068
[6] N. Apreutesei, A. Ducrot, and V. Volpert, “Travelling waves for integro-differential equations in population dynamics,” Discrete and Continuous Dynamical Systems. Series B, vol. 11, no. 3, pp. 541-561, 2009. · Zbl 1173.35541 · doi:10.3934/dcdsb.2009.11.541
[7] H. Berestycki, G. Nadin, B. Perthame, and L. Ryzhik, “The non-local Fisher-KPP equation: travelling waves and steady states,” Nonlinearity, vol. 22, no. 12, pp. 2813-2844, 2009. · Zbl 1195.35088 · doi:10.1088/0951-7715/22/12/002
[8] N. F. Britton, “Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,” SIAM Journal on Applied Mathematics, vol. 50, no. 6, pp. 1663-1688, 1990. · Zbl 0723.92019 · doi:10.1137/0150099
[9] J. F. Collet and V. A. Volpert, “Computation of the index of linear elliptic operators in unbounded cylinders,” Journal of Functional Analysis, vol. 164, no. 1, pp. 34-59, 1999. · Zbl 0930.35057 · doi:10.1006/jfan.1999.3392
[10] I. Demin and V. Volpert, “Existence of waves for a nonlocal reaction-diffusion equation,” Mathematical Modelling of Natural Phenomena, vol. 5, no. 5, pp. 80-101, 2010. · Zbl 1232.35181 · doi:10.1051/mmnp/20105506
[11] A. Ducrot, “Travelling wave solutions for a scalar age-structured equation,” Discrete and Continuous Dynamical Systems. Series B, vol. 7, no. 2, pp. 251-273, 2007. · Zbl 1195.35090 · doi:10.3934/dcdsb.2007.7.251
[12] I. C. Gohberg and M. G. Kreĭn, “The basic propositions on defect numbers, root numbers and indices of linear operators,” American Mathematical Society Translations, vol. 13, pp. 185-264, 1960. · Zbl 0089.32201
[13] S. A. Gourley, “Travelling front solutions of a nonlocal Fisher equation,” Journal of Mathematical Biology, vol. 41, no. 3, pp. 272-284, 2000. · Zbl 0982.92028 · doi:10.1007/s002850000047
[14] M. A. Krasnoselskii and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, vol. 263, Springer, Berlin, Germany, 1984.
[15] R. Lefever and O. Lejeune, “On the origin of tiger bush,” Bulletin of Mathematical Biology, vol. 59, no. 2, pp. 263-294, 1997. · Zbl 0903.92031 · doi:10.1007/BF02462004
[16] B. Perthame and S. Génieys, “Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit,” Mathematical Modelling of Natural Phenomena, vol. 2, no. 4, pp. 135-151, 2007. · Zbl 1337.35077 · doi:10.1051/mmnp:2008029
[17] A. Volpert and V. Volpert, “The Fredholm property of elliptic operators in unbounded domains,” Transactions of the Moscow Mathematical Society, vol. 67, pp. 127-197, 2006. · Zbl 1219.35078
[18] V. Volpert and A. Volpert, “Properness and topological degree for general elliptic operators,” Abstract and Applied Analysis, vol. 2003, no. 3, pp. 129-181, 2003. · Zbl 1030.35038 · doi:10.1155/S1085337503204073
[19] V. A. Volpert, A. I. Volpert, and J. F. Collet, “Topological degree for elliptic operators in unbounded cylinders,” Advances in Differential Equations, vol. 4, no. 6, pp. 777-812, 1999. · Zbl 0952.35038
[20] V. Volpert, “Elliptic partial differential equations,” in Fredholm Theory of Elliptic Problems in Unbounded Domains, vol. 1, Birkhäauser, Boston, Mass, USA, 2011. · Zbl 1222.35002
[21] A. I. Volpert, V. Volpert, and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, vol. 140 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1994. · Zbl 0805.35143
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.