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

Mathematical Modelling of Natural Phenomena (2009)

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

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topHuang, 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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- W. Huang. Monotonicity of heteroclinic orbits and spectral properties of variational equations for delay differential equations. J. Diff. Equations, 162 (2000), 91–139.
- 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).
- 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.
- 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.
- J. Wu. Theory and applications of partial functional differential equations. Applied Mathematical Science, Vol. 119, Springer, Berlin, 1996.
- 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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