Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 561 - 569, Volume 23, Issue 1/2, January/February 2009      doi:10.3934/dcds.2009.23.561

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Jonathan J. Wylie - Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China (email)
Huaxiong Huang - Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada (email)
Robert M. Miura - Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, United States (email)

Abstract: 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:  Diffusion equations with degenerate nonlinear sources, linear stability, traveling waves, applications
Mathematics Subject Classification:  Primary: 35K57, 35B35, 74J30; Secondary: 92C20, 92D30

Received: December 2007;      Revised: April 2008;      Available Online: September 2008.