# American Institute of Mathematical Sciences

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

## Existence and longtime behavior of a biofilm model

 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.
Citation: Messoud A. Efendiev, Sergey Zelik, Hermann J. Eberl. Existence and longtime behavior of a biofilm model. Communications on Pure & Applied Analysis, 2009, 8 (2) : 509-531. doi: 10.3934/cpaa.2009.8.509
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