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

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January & February 2009, 23(1&2): 561-569. doi: 10.3934/dcds.2009.23.561

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

Jonathan J. Wylie ^1, , Robert M. Miura ^3, and Huaxiong Huang ^2,

1.	Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China
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, United States

Received December 2007 Revised April 2008 Published September 2008

Abstract
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Diffusion equations with degenerate nonlinear source terms arise in many different applications, e.g., in the theory of epidemics, in models of cortical spreading depression, and in models of evaporation and condensation in porous media. In this paper, we consider 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 of uniform rest states and show that traveling wave trajectories connecting two stable rest states can exist generically only for discrete wave speeds. Furthermore, we show that families of traveling waves with a continuum of wave speeds cannot exist.

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

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

Citation: Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561

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