Infinite Dimensional Dynamical Systems pp 421-443 | Cite as
Traveling Wavefronts in Lattice Differential Equations with Time Delay and Global Interaction
Chapter
First Online:
Abstract
In this paper, we study the existence of traveling wave solutions in lattice differential equations with time delay and global interaction Following an idea in [10], we are able to relate the existence of traveling wavefront to the existence of heteroclinic connecting orbits of the corresponding ordinary delay differential equations
$$\begin{array}{rcl}{ u^{\prime}}_{n}(t)& =& D\sum\limits_{i\in {\mathit{Z}}^{q}\setminus \{0\}}J(i)[{u}_{n-i}(t) - {u}_{n}(t)] \\ & & +F\left ({u}_{n}(t),\sum\limits_{i\in {\mathit{Z}}^{q}}K(i){\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g({u}_{ n-i}(t + \theta ))\right )\end{array}$$
$$u^{\prime}(t) = F\left (u(t),{\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g(u(t + \theta ))\right ).$$
Notes
Acknowledgments
Research partially supported by the National Natural Science Foundation of China (SM), by Natural Sciences and Engineering Research Council of Canada, and by a Premier Research Excellence Award of Ontario (XZ)
Received 2/20/2009; Accepted 6/30/2010
References
- 1.P.W. Bates, A. Chmaj, A discrete convolution model for phase transitions. Arch. Rational Mech. Anal. 150, 281–305 (1999)MathSciNetCrossRefGoogle Scholar
- 2.P.W. Bates, P.C. Fife, X.F. Ren, X.F. Wang, Traveling waves in a convolution model for phase transitions. Arch. Rational Mech. Anal. 138, 105–136 (1997)MathSciNetCrossRefGoogle Scholar
- 3.J.W. Cahn, J. Mallet-Paret, E.S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59, 455–493 (1999)MathSciNetzbMATHGoogle Scholar
- 4.J. Carr and Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132, 2433–2439 (2004)MathSciNetCrossRefGoogle Scholar
- 5.X. Chen, J.S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J. Differ. Equat. 184, 549–569 (2002)MathSciNetCrossRefGoogle Scholar
- 6.X. Chen, J.S. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics. Math. Ann. 326, 123–146 (2003)MathSciNetCrossRefGoogle Scholar
- 7.S.-N. Chow, J.K. Hale, Methods of Bifurcation Theory (Springer, New York, 1982)CrossRefGoogle Scholar
- 8.S.-N. Chow, J. Mallet-Paret, W. Shen, Traveling waves in lattice dynamical systems. J. Differ. Equat. 149, 248–291 (1998)MathSciNetCrossRefGoogle Scholar
- 9.N. Dance, P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew Math. 419, 125–139 (1991)MathSciNetGoogle Scholar
- 10.T. Faria, W.Z. Huang, J.H. Wu, Traveling waves for delayed reaction-diffusion equations with global response. Proc. Roy. Soc. Lond. A. 462, 229–261 (2006)CrossRefGoogle Scholar
- 11.J. Keener, Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 22, 556–572 (1987)MathSciNetCrossRefGoogle Scholar
- 12.S. Ma, X. Liao, J. Wu, Traveling wave solutions for planner lattice differential euations with applications to neural networks. J. Differ. Equat. 182, 269–297 (2002)CrossRefGoogle Scholar
- 13.S. Ma, P. Weng, X. Zou, Asymptotic speed of propagation and traveling wavefront in a non-local delayed lattice differential equation. Nonl. Anal. TMA. 65, 1858–1890 (2006)MathSciNetCrossRefGoogle Scholar
- 14.S. Ma, X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay. J. Differ. Equat. 217, 54–87 (2005)CrossRefGoogle Scholar
- 15.S. Ma, X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction. J. Differ. Equat. 212, 129–190 (2005)MathSciNetCrossRefGoogle Scholar
- 16.J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems. J. Dynam. Differ. Equat. 11, 49–127 (1999)MathSciNetCrossRefGoogle Scholar
- 17.H.L. Smith, Invariant curves for mappings. SIAM J. Math. Anal. 17, 1053–1067 (1986)MathSciNetCrossRefGoogle Scholar
- 18.H.L. Smith, Monotone Dynamical Systems, an introduction to the theory of competitive and cooperative system, Mathematical Surveys and Monographs, vol. 11 (Amer. Math. Soc., Providence, 1995)Google Scholar
- 19.H.L. Smith, H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional differential equations. J. Math. Anal. Appl. 21, 637–692 (1990)MathSciNetzbMATHGoogle Scholar
- 20.H.L. Smith, H. Thieme, Strongly order preserving semiflows generated by functional differential equations. J. Differ. Equat. 93, 332–363 (1991)MathSciNetCrossRefGoogle Scholar
- 21.J.W.-H. So, J. Wu, X. Zou, A reaction-diffusion model for a single species with age structure. I Traveling wavefronts on unbounded domains. Proc. R. Soc. Lond. A. 457, 1841–1853 (2001)CrossRefGoogle Scholar
- 22.P.X. Weng, H.X. Huang, J.H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. 68, 409–439 (2003)MathSciNetCrossRefGoogle Scholar
- 23.J. Wu, H. Freedman, R. Miller, Heteroclinic orbits and convergence order-preserving set-condensing semiflows with applications to integro-differential equations. J. Integral Equ. Appl. 7, 115–133 (1995)CrossRefGoogle Scholar
- 24.J. Wu, X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Differ. Equat. 135, 315–357 (1997)MathSciNetCrossRefGoogle Scholar
- 25.X. Zou, Traveling wave fronts in spatially discrete reaction-diffusion equations on higher dimensional lattices, in Differential Equations and Computational Simulations III, ed. by J. Graef et al., Electronic Journal of Differential Equations, Conference 01 (1997), pp. 211–222Google Scholar
Copyright information
© Springer Science+Business Media New York 2013