Uniqueness of Monotone Mono-stable Waves for Reaction-Diffusion Equations with Time Delay

W. Huang; M. Han; M. Puckett

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 2, page 48-67
  • ISSN: 0973-5348

Abstract

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Many models in biology and ecology can be described by reaction-diffusion equations wit time delay. One of important solutions for these type of equations is the traveling wave solution that shows the phenomenon of wave propagation. The existence of traveling wave fronts has been proved for large class of equations, in particular, the monotone systems, such as the cooperative systems and some competition systems. However, the problem on the uniqueness of traveling wave (for a fixed wave speed) remains unsolved. In this paper, we show that, for a class of monotone diffusion systems with time delayed reaction term, the mono-stable traveling wave font is unique whenever it exists.

How to cite

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Huang, W., Han, M., and Puckett, M.. "Uniqueness of Monotone Mono-stable Waves for Reaction-Diffusion Equations with Time Delay." Mathematical Modelling of Natural Phenomena 4.2 (2009): 48-67. <http://eudml.org/doc/222382>.

@article{Huang2009,
abstract = { Many models in biology and ecology can be described by reaction-diffusion equations wit time delay. One of important solutions for these type of equations is the traveling wave solution that shows the phenomenon of wave propagation. The existence of traveling wave fronts has been proved for large class of equations, in particular, the monotone systems, such as the cooperative systems and some competition systems. However, the problem on the uniqueness of traveling wave (for a fixed wave speed) remains unsolved. In this paper, we show that, for a class of monotone diffusion systems with time delayed reaction term, the mono-stable traveling wave font is unique whenever it exists. },
author = {Huang, W., Han, M., Puckett, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {reaction-diffusion equations; time delay; traveling waves; monotone systems; monotone systems},
language = {eng},
month = {3},
number = {2},
pages = {48-67},
publisher = {EDP Sciences},
title = {Uniqueness of Monotone Mono-stable Waves for Reaction-Diffusion Equations with Time Delay},
url = {http://eudml.org/doc/222382},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Huang, W.
AU - Han, M.
AU - Puckett, M.
TI - Uniqueness of Monotone Mono-stable Waves for Reaction-Diffusion Equations with Time Delay
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/3//
PB - EDP Sciences
VL - 4
IS - 2
SP - 48
EP - 67
AB - Many models in biology and ecology can be described by reaction-diffusion equations wit time delay. One of important solutions for these type of equations is the traveling wave solution that shows the phenomenon of wave propagation. The existence of traveling wave fronts has been proved for large class of equations, in particular, the monotone systems, such as the cooperative systems and some competition systems. However, the problem on the uniqueness of traveling wave (for a fixed wave speed) remains unsolved. In this paper, we show that, for a class of monotone diffusion systems with time delayed reaction term, the mono-stable traveling wave font is unique whenever it exists.
LA - eng
KW - reaction-diffusion equations; time delay; traveling waves; monotone systems; monotone systems
UR - http://eudml.org/doc/222382
ER -

References

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  1. A. Berman, R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics 9, SIAM, Philadelphia, 1994.  
  2. J. Carr, A. Chmaj. Uniqueness of travelling waves for nonlocal monostable equations. Proc. Amer. Math. Soc., 132 (2004), 2433–2439.  
  3. T. Faria, W. Huang, J. Wu. Traveling waves for delayed reaction-diffusion equations with global response. Proc. Royal Society A, 462 (2006), 229-261.  
  4. W. Huang. Monotonicity of heteroclinic orbits and spectral properties of variational equations for delay differential equations. J. Diff. Equations, 162 (2000), 91–139.  
  5. W. Huang, M. Pucket. A note on uniqueness of monotone mono-stable waves for reaction-diffusion equations. Inter. J. Qualitative Theory of Diff. Equations and App., in press (2008).  
  6. H. Thieme, X-Q. Zhao. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J. Diff. Equations, 195(2003), 430–470.  
  7. V.I. Volpert, V.A. Volpert, V.A. Volpert. Traveling wave solutions of parabolic systems. Translations of Math. Monographs, 140, Amer. Math. Soc., Providence, 1994.  
  8. J. Wu. Theory and applications of partial functional differential equations. Applied Mathematical Science, Vol. 119, Springer, Berlin, 1996.  
  9. J. Wu, X. Zou. Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method. Proc. Amer. Math. Soc., 125 (1997) 2589-2598.  

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