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Traveling Wavefronts in Lattice Differential Equations with Time Delay and Global Interaction

  • Shiwang Ma
  • Xingfu Zou
Chapter
First Online:
Part of the Fields Institute Communications book series (FIC, volume 64)

Abstract

In this paper, we study the existence of traveling wave solutions in lattice differential equations with time delay and global interaction
$$\begin{array}{rcl}{ u^{\prime}}_{n}(t)& =& D\sum\limits_{i\in {\mathit{Z}}^{q}\setminus \{0\}}J(i)[{u}_{n-i}(t) - {u}_{n}(t)] \\ & & +F\left ({u}_{n}(t),\sum\limits_{i\in {\mathit{Z}}^{q}}K(i){\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g({u}_{ n-i}(t + \theta ))\right )\end{array}$$
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
$$u^{\prime}(t) = F\left (u(t),{\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g(u(t + \theta ))\right ).$$
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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. 1.
    P.W. Bates, A. Chmaj, A discrete convolution model for phase transitions. Arch. Rational Mech. Anal. 150, 281–305 (1999)MathSciNetCrossRefGoogle Scholar
  2. 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. 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. 4.
    J. Carr and Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132, 2433–2439 (2004)MathSciNetCrossRefGoogle Scholar
  5. 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. 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. 7.
    S.-N. Chow, J.K. Hale, Methods of Bifurcation Theory (Springer, New York, 1982)CrossRefGoogle Scholar
  8. 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. 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. 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. 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. 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. 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. 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. 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. 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. 17.
    H.L. Smith, Invariant curves for mappings. SIAM J. Math. Anal. 17, 1053–1067 (1986)MathSciNetCrossRefGoogle Scholar
  18. 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. 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. 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. 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. 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. 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. 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. 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

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinP. R. China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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