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March  2009, 8(2): 509-531. doi: 10.3934/cpaa.2009.8.509

Existence and longtime behavior of a biofilm model

Messoud A. Efendiev 1, , Sergey Zelik 2, and  Hermann J. Eberl 3,

1. 

Institute of Biomathematics and Biometry, HelmholtzZentrum München, Ingolstädter Landstrasse 1, 85764 Neuherberg, Germany
2. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH
3. 

Department of Mathematics and Statistics, University of Guelph, Guelph, On, N1G 2W1, Canada

Received  March 2008 Revised  August 2008 Published  December 2008

A nonlinear, density-dependent system of diffusion-reaction equations describing development of bacterial biofilms is analyzed. It comprises two non-standard diffusion effects, degeneracy as in the porous medium equation and fast diffusion. The existence of a unique bounded solution and a global attractor is proved in dependence of the boundary conditions. This is achieved by studying a system of non-degenerate auxiliary approximation equations and the construction of a Lipschitz continuous semigroup by passing to the limit in the approximation parameter. Numerical examples are included in order to illustrate the main result.
Keywords: biofilm, global attractor, Nonlinear diffusion, degeneracy.
Mathematics Subject Classification: Primary: 35K65; Secondary: 92D25.
Citation: Messoud A. Efendiev, Sergey Zelik, Hermann J. Eberl. Existence and longtime behavior of a biofilm model. Communications on Pure and Applied Analysis, 2009, 8 (2) : 509-531. doi: 10.3934/cpaa.2009.8.509
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