Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves

Jonathan J. Wylie; Robert M. Miura; Huaxiong Huang

doi:10.3934/dcds.2009.23.561

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2009, Volume 23, Issue 1&2: 561-569. Doi: 10.3934/dcds.2009.23.561

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Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves

1.
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong
2.
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3
3.
Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102

Received: November 30, 2007

Revised: March 31, 2008

Published: April 2009

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Abstract

Diffusion equations with degenerate nonlinear source terms arisein many different applications, e.g., in the theory of epidemics,in models of cortical spreading depression, and in models ofevaporation and condensation in porous media. In this paper, weconsider a generalization of these models to a system of $n$coupled diffusion equations with identical nonlinear source terms.We determine simple conditions that ensure the linear stability ofuniform rest states and show that traveling wave trajectoriesconnecting two stable rest states can exist generically only fordiscrete wave speeds. Furthermore, we show that families oftraveling waves with a continuum of wave speeds cannot exist.

Keywords:

Diffusion equations with degenerate nonlinear sources,
linear stability,
traveling waves,
applications.

Mathematics Subject Classification: Primary: 35K57, 35B35, 74J30; Secondary: 92C20, 92D30.

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