Traveling wave front for partial neutral differential equations
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- by Eduardo Hernández and Jianhong Wu PDF
- Proc. Amer. Math. Soc. 146 (2018), 1603-1617 Request permission
Abstract:
By using Schauder’s point fixed theorem we study the existence of a traveling wave front for reaction-diffusion differential equations of the neutral type. Some examples arising in populations dynamics are presented.References
- N. F. Britton, Reaction-diffusion equations and their applications to biology, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. MR 866143
- Hui Fang and Jibin Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl. 259 (2001), no. 1, 8–17. MR 1836440, DOI 10.1006/jmaa.2000.7340
- Paul C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, Berlin-New York, 1979. MR 527914
- R. A. Fisher, The wave of advance of advantageous gene, Ann. Eugen. 7 (1937) 355–369
- H. I. Freedman and Yang Kuang, Some global qualitative analyses of a single species neutral delay differential population model, Rocky Mountain J. Math. 25 (1995), no. 1, 201–215. Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1992). MR 1340003, DOI 10.1216/rmjm/1181072278
- K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1163190, DOI 10.1007/978-94-015-7920-9
- Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI 10.1007/BF00281373
- Jack K. Hale, Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl. 39 (1994), no. 4, 339–344. MR 1317773
- Warren M. Hirsch, Herman Hanisch, and Jean-Pierre Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math. 38 (1985), no. 6, 733–753. MR 812345, DOI 10.1002/cpa.3160380607
- A. Kolmogorov, I. Petrovskii, and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh. 1 (1937) 1–26.
- Yang Kuang and Alan Feldstein, Boundedness of solutions of a nonlinear nonautonomous neutral delay equation, J. Math. Anal. Appl. 156 (1991), no. 1, 293–304. MR 1102613, DOI 10.1016/0022-247X(91)90398-J
- Yang Kuang, Global stability in one or two species neutral delay population models, Canad. Appl. Math. Quart. 1 (1993), no. 1, 23–45. MR 1226768
- Yang Kuang, Qualitative analysis of one- or two-species neutral delay population models, SIAM J. Math. Anal. 23 (1992), no. 1, 181–200. MR 1145167, DOI 10.1137/0523009
- Yubin Liu and Peixuan Weng, Asymptotic pattern for a partial neutral functional differential equation, J. Differential Equations 258 (2015), no. 11, 3688–3741. MR 3322982, DOI 10.1016/j.jde.2015.01.016
- Alessandra Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal. 21 (1990), no. 5, 1213–1224. MR 1062400, DOI 10.1137/0521066
- Shiwang Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations 171 (2001), no. 2, 294–314. MR 1818651, DOI 10.1006/jdeq.2000.3846
- J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836, DOI 10.1007/978-3-662-08539-4
- Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. MR 295683, DOI 10.1090/S0033-569X-1971-0295683-6
- Klaus W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc. 302 (1987), no. 2, 587–615. MR 891637, DOI 10.1090/S0002-9947-1987-0891637-2
- Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. MR 1297766, DOI 10.1090/mmono/140
- Jianhong Wu and Xingfu Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001), no. 3, 651–687. MR 1845097, DOI 10.1023/A:1016690424892
- Jianhong Wu and Xingfu Zou, Erratum to: “Traveling wave fronts of reaction-diffusion systems with delay” [J. Dynam. Differential Equations 13 (2001), no. 3, 651–687; MR1845097], J. Dynam. Differential Equations 20 (2008), no. 2, 531–533. MR 2385719, DOI 10.1007/s10884-007-9090-1
- Jianhong Wu and Huaxing Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations 124 (1996), no. 1, 247–278. MR 1368068, DOI 10.1006/jdeq.1996.0009
- J. Wu and H. Xia, Rotating waves in neutral partial functional-differential equations, J. Dynam. Differential Equations 11 (1999), no. 2, 209–238. MR 1695243, DOI 10.1023/A:1021973228398
- Xingfu Zou and Jianhong Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2589–2598. MR 1415345, DOI 10.1090/S0002-9939-97-04080-X
Additional Information
- Eduardo Hernández
- Affiliation: Departamento de Computação e Matemática, Faculdade de Filosofia Ciencias e Letras de Ribeirão Preto Universidade de São Paulo, CEP 14040-901 Ribeirão Preto, SP, Brazil
- Email: lalohm@ffclrp.usp.br
- Jianhong Wu
- Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
- MR Author ID: 226643
- Email: wujh@mathstat.yorku.ca
- Received by editor(s): March 17, 2017
- Received by editor(s) in revised form: May 4, 2017, and May 12, 2017
- Published electronically: November 7, 2017
- Additional Notes: The work of the first author was supported by Fapesp Grant 2014/25818-9 and by the Natural Sceinces and Engineering Research Council of Canada. This work was developed during the first author’s visit to York University
- Communicated by: Wenxian Shen
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1603-1617
- MSC (2010): Primary 35K57, 35C07, 34K40
- DOI: https://doi.org/10.1090/proc/13824
- MathSciNet review: 3754345