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Traveling waves for nonlocal delayed diffusion equations via auxiliary equations. (English) Zbl 1114.34061
Summary: We study the existence of traveling wave solutions for a class of delayed nonlocal reaction-diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction-diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder’s fixed point theorem, we then show that there exists a constant \(c_{*} > 0\) such that for each \(c > c_{*}\), the equation under consideration admits a traveling wavefront solution with speed \(c\), which is not necessary to be monotonic.

34K30 Functional-differential equations in abstract spaces
35B40 Asymptotic behavior of solutions to PDEs
35R10 Functional partial differential equations
34K10 Boundary value problems for functional-differential equations
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[1] Al-Omari, J.; Gourley, S.A., Monotone travelling fronts in an age-structured reaction – diffusion model of a single species, J. math. biol., 45, 294-312, (2002) · Zbl 1013.92032
[2] Chen, X.; Guo, J.S., Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. differential equations, 184, 549-569, (2002) · Zbl 1010.39004
[3] Faria, T.; Huang, W.; Wu, J., Traveling waves for delayed reaction – diffusion equations with global response, Proc. R. soc. lond. ser. A, 462, 2065, 229-261, (2006) · Zbl 1149.35368
[4] Gourley, S.A.; Kuang, Y., Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. soc. lond. ser. A, 459, 2034, 1563-1579, (2003) · Zbl 1047.92037
[5] S.A. Gourley, J.W.-H. So, J. Wu, Non-locality of reaction – diffusion equations induced by delay: Biological modeling and nonlinear dynamics, preprint · Zbl 1128.35360
[6] Huang, J.; Zou, X., Existence of traveling wavefronts of delayed reaction – diffusion systems without monotonicity, Discrete contin. dyn. syst. ser. A, 9, 925-936, (2003) · Zbl 1093.34030
[7] Ma, S.W., Traveling wavefronts for delayed reaction – diffusion systems via a fixed point theorem, J. differential equations, 171, 294-314, (2001) · Zbl 0988.34053
[8] S.W. Ma, J.H. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Differential Equations (2006), in press, DOI: 10.1007/s10884-006-9065-7
[9] So, J.W.-H.; Wu, J.H.; Zou, X.F., A reaction – diffusion model for a single species with age structure. I. traveling wavefronts on unbounded domains, Proc. R. soc. lond. ser. A, 457, 1841-1853, (2001) · Zbl 0999.92029
[10] Thieme, H.R.; Zhao, X.-Q., Asymptotic speed of spread and traveling waves for integral equations and delayed reaction – diffusion models, J. differential equations, 195, 430-470, (2003) · Zbl 1045.45009
[11] Volpert, A.I.; Volpert, V.A.; Volpert, V.A., Traveling wave solutions of parabolic systems, Transl. math. monogr., vol. 140, (1994), Amer. Math. Soc. Providence, RI · Zbl 0835.35048
[12] Wu, J.; Zou, X., Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. differential equations, 135, 315-357, (1997) · Zbl 0877.34046
[13] Wu, J.; Zou, X., Traveling wave fronts of reaction – diffusion systems with delay, J. dynam. differential equations, 13, 651-687, (2001) · Zbl 0996.34053
[14] Wu, J.H., Theory and applications of partial functional-differential equations, Appl. math. sci., vol. 119, (1996), Springer-Verlag New York
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