Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

Journal of Differential Equations

Volumn 217, Issue 1, 2005, Pages 54-87

  Ma, Shiwang ^a,d   Zou, Xingfu ^b,c  

^a SHANGHAI JIAO TONG UNIVERSITY   (China)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

^c UNIVERSITY OF WESTERN ONTARIO   (Canada)

^d NANKAI UNIVERSITY   (China)

Author keywords

Asymptotic stability; Delay; Discrete reaction diffusion equation; Existence; Initial value problem; Monostable; Travelling wavefront; Uniqueness

Indexed keywords

EID: 24644479145     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2005.05.004     Document Type: Article

References (29)

1. P.W. Bates X. Chen A. Chmaj Travelling waves of bistable dynamics on a lattice SIAM J. Math. Anal. 35 2003 520-546
2. P.W. Bates A. Chmaj A discrete convolution model for phase transitions Arch. Rational Mech. Anal. 150 1999 281-305
3. J. Bell C. Cosner Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons Quart. Appl. Math. 42 1984 1-14
4. J. Cahn J. Mallet-Paret E.S. Van Vleck Travelling wave solutions for systems of ODEs on a two-dimensional spatial lattice SIAM J. Appl. Math. 59 1999 455-493
5. J. Carr, A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004) 2433-2439.
6. X. Chen Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations Adv. Differential Equations 2 1997 125-160
7. X. Chen J.S. Guo Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations J. Differential Equations 184 2002 549-569
8. X. Chen J.S. Guo Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics Math. Ann. 326 2003 123-146
9. S.-N. Chow J. Mallet-Paret W. Shen Travelling waves in lattice dynamical systems J. Differential Equations 149 1998 248-291
10. T. Faria, W. Huang, J. Wu, Travelling waves for delayed reaction-diffusion equations with non-local response, preprint.
11. C.-H. Hsu S.-S. Lin Existence and multiplicity of traveling waves in a lattice dynamical system J. Differential Equations 164 2000 431-450
12. J.P. Keener Propagation and its failure to coupled systems of discrete excitable cells SIAM J. Appl. Math. 47 1987 556-572
13. S.W. Ma X.X. Liao J. Wu Traveling wave solutions for planar lattice differential systems with applications to neural networks J. Differential Equations 182 2002 269-297
14. J. Mallet-Paret The global structure of traveling waves in spatially discrete dynamical systems J. Dynam. Differential Equations 11 1999 49-127
15. K.W. Schaaf Asymptotic behavior and traveling wave solutions for parabolic functional differential equations Trans. Amer. Math. Soc. 302 1987 587-605
16. W. Shen Travelling waves in time almost periodic structures governed by bistable nonlinearities. II. Existence J. Differential Equations 159 1999 55-101
17. W. Shen Travelling waves in time almost periodic structures governed by bistable nonlinearities. II. Stability and uniqueness J. Differential Equations 159 1999 1-54
18. B. Shorrocks I.R. Swingland Living in a Patch Environment 1990 Oxford University Press New York
19. H.L. Smith X.Q. Zhao Global asymptotic stability of travelling waves in delayed reaction-diffusion equations SIAM J. Math. Anal. 31 2000 514-534
20. J.W.-H. So J. Wu X. Zou A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains Proc. Roy. Soc. London A 457 2001 1841-1853
21. J.W.-H. So Y. Yang Dirichlet problem for the diffusive Nicholson's blowflies equation J. Differential Equations 150 1998 317-348
22. P.X. Weng H. Huang J. Wu Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction IMA J. Appl. Math. 68 2003 409-439
23. J. Wu X. Zou Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations J. Differential Equations 135 1997 315-357
24. Y. Yang, J.W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation, Dynamical Systems and Differential Equations, vol. II, Springfield, MO, 1996: Discrete Contin. Dynam. Systems Added Volume II (1998) 333-352.
25. B. Zinner Stability of travelling wavefronts for the discrete Nagumo equation SIAM J. Math. Anal. 22 1991 1016-1020
26. B. Zinner Existence of travelling wavefront solution for the discrete Nagumo equation J. Differential Equations 96 1992 1-27
27. B. Zinner G. Harris W. Hudson Travelling wavefronts for the discrete Fisher's equation J. Differential Equations 105 1993 46-62
28. X. Zou, Traveling wave fronts in spatially discrete reaction-diffusion equations on higher-dimensional lattices, in: Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations, Mississippi State, MS, 1997; Electron. J. Differential Equations, Conf. 1. (electronic), 211-221.
29. X. Zou Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type J. Comput. Appl. Math. 146 2002 309-321