Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T16:40:11.634Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  26 March 2009

W. Huang ,
M. Han  and
M. Puckett
W. Huang*
Affiliation:
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA
M. Han
Affiliation:
Department of Mathematics, Sanghai Normal University Shangai, China
M. Puckett
Affiliation:
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA
*
[email protected]
Get access

Abstract

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

Keywords

reaction-diffusion equationstime delaytraveling wavesmonotonesystems
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 2: Delay equations in biology , 2009 , pp. 48 - 67
Copyright
© EDP Sciences, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Berman, R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics 9, SIAM, Philadelphia, 1994.
Carr, J., Chmaj, A.. Uniqueness of travelling waves for nonlocal monostable equations. Proc. Amer. Math. Soc., 132 (2004), 2433–2439. CrossRef
Faria, T., Huang, W., Wu, J.. Traveling waves for delayed reaction-diffusion equations with global response. Proc. Royal Society A, 462 (2006), 229-261. CrossRef
Huang, W.. Monotonicity of heteroclinic orbits and spectral properties of variational equations for delay differential equations. J. Diff. Equations, 162 (2000), 91–139. CrossRef
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.