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On the existence of traveling waves for delayed reaction-diffusion equations. (English) Zbl 1167.35023
Summary: We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.

MSC:
35K57Reaction-diffusion equations
34K30Functional-differential equations in abstract spaces
35R10Partial functional-differential equations
45K05Integro-partial differential equations
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References:
[1] Al-Omari, J.; Gourley, S.: Monotone travelling fronts in an age-structured reaction -- diffusion model of a single species, J. math. Biol. 45, 294-312 (2002) · Zbl 1013.92032 · doi:10.1007/s002850200159
[2] Boumenir, A.; Nguyen, V.: Perron theorem in the monotone iteration method for traveling waves in delayed reaction -- diffusion equations, J. differential equations 244, 1551-1570 (2008) · Zbl 1154.34031 · doi:10.1016/j.jde.2008.01.004
[3] Diekmann, O.: Thresholds and travelling waves for the geographical spread of an infection, J. math. Biol. 6, 109-130 (1978) · Zbl 0415.92020 · doi:10.1007/BF02450783
[4] Faria, T.; Trofimchuk, S.: Nonmonotone travelling waves in a single species reaction -- diffusion equation with delay, J. differential equations 228, 357-376 (2006) · Zbl 1217.35102 · doi:10.1016/j.jde.2006.05.006
[5] Faria, T.; Huang, W.; Wu, J.: Traveling waves for delayed reaction -- diffusion equations with global response, Proc. R. Soc. lond. Ser. A math. Phys. eng. Sci. 462, 229-261 (2006) · Zbl 1149.35368 · doi:10.1098/rspa.2005.1554
[6] Gourley, S. A.: Travelling front solutions of a nonlocal Fisher equation, J. math. Biol. 41, 272-284 (2000) · Zbl 0982.92028 · doi:10.1007/s002850000047
[7] Gourley, S. A.; Kuang, Y.: Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. lond. Ser. A math. Phys. eng. Sci. 459, 1563-1579 (2003) · Zbl 1047.92037 · doi:10.1098/rspa.2002.1094
[8] Hsu, S.; Zhao, X. -Q.: Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. anal. 40, 776-789 (2008) · Zbl 1160.37031 · doi:10.1137/070703016
[9] Li, B.; Lewis, M.; Weinberger, H.: Existence of traveling waves for integral recursions with nonmonotone growth functions, J. math. Biol. 58, 323-338 (2009) · Zbl 1162.92030 · doi:10.1007/s00285-008-0175-1
[10] Ma, S.: Traveling wavefronts for delayed reaction -- diffusion systems via a fixed point theorem, J. differential equations 171, 294-314 (2001) · Zbl 0988.34053 · doi:10.1006/jdeq.2000.3846
[11] Ma, S.: Traveling waves for non-local delayed diffusion equations via auxiliary equation, J. differential equations 237, 259-277 (2007) · Zbl 1114.34061 · doi:10.1016/j.jde.2007.03.014
[12] Murray, J. D.: Mathematical biology, (1989) · Zbl 0682.92001
[13] Schaaf, K. W.: Asymptotic behavior and travelling wave solutions for parabolic functional differential equations, Trans. amer. Math. soc. 302, 587-615 (1987) · Zbl 0637.35082 · doi:10.2307/2000859
[14] So, J.; Wu, J.; Zou, X.: A reaction -- diffusion model for a single species with age structure. I. traveling wavefronts on unbounded domains, Proc. R. Soc. lond. Ser. A math. Phys. eng. Sci. 457, 1841-1853 (2001) · Zbl 0999.92029 · doi:10.1098/rspa.2001.0789
[15] Thieme, H.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. math. Biol. 8, 173-187 (1979) · Zbl 0417.92022 · doi:10.1007/BF00279720
[16] Thieme, H.; Zhao, X.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction -- diffusion models, J. differential equations 195, 430-470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X
[17] Wang, Z.; Li, W.; Ruan, S.: Travelling wave fronts in reaction -- diffusion systems with spatio-temporal delays, J. differential equations 222, 185-232 (2006) · Zbl 1100.35050 · doi:10.1016/j.jde.2005.08.010
[18] Weinberger, H. F.: Asymptotic behavior of a model in population genetics, Lecture notes in math. 648, 47-96 (1978) · Zbl 0383.35034
[19] Wu, J.; Zou, X.: Traveling wave fronts of reaction -- diffusion systems with delay, J. dynam. Differential equations 13, 651-687 (2001) · Zbl 0996.34053 · doi:10.1023/A:1016690424892
[20] Wu, J.; Zou, X.: Erratum to ”traveling wave fronts of reaction -- diffusion systems with delays” [J. Dynam. differential equations 13 (2001) 651 -- 687], J. dynam. Differential equations 20, 531-533 (2008) · Zbl 0996.34053
[21] Wu, J.: Theory and applications of partial functional-differential equations, Appl. math. Sci. 119 (1996) · Zbl 0870.35116
[22] Zhao, X.; Wang, W.: Fisher waves in an epidemic model, Discrete contin. Dyn. syst. Ser. B 4, 1117-1128 (2004) · Zbl 1097.34022 · doi:10.3934/dcdsb.2004.4.1117