COMMUNICATIONS ON doi:10.3934/cpaa.2009.8.509 PURE AND APPLIED ANALYSIS Volume 8, Number 2, March 2009 pp. 509–531 EXISTENCE AND LONGTIME BEHAVIOR OF A BIOFILM MODEL Messoud A. Efendiev Institute of Biomathematics and Biometry, HelmholtzZentrum M¨ unchen Ingolst¨ adter Landstrasse 1, 85764 Neuherberg, Germany Sergey V. Zelik Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK Hermann J. Eberl Department of Mathematics and Statistics University of Guelph, Guelph, On, N1G 2W1, Canada (Communicated by Alain Miranville) Abstract. A nonlinear, density-dependent system of diffusion-reaction equa- tions 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. 1. Introduction. Biofilms play a very important role in many scientific and tech- nological areas. Consequently, they are studied in many disciplines and biofilm research is a truly interdisciplinary research topic. Biofilms are the most success- ful life form on earth growing virtually everywhere, where nutrients are available to feed bacteria. In fact, most bacteria live in biofilm colonies and only a small minority appears as suspended planktonic organisms. Biofouling, biocorrosion, and bacterial infections are harmful impacts of biofilms. On the other hand, beneficial properties of biofilms are used in environmental engineering for wastewater treat- ment, groundwater protection, and soil remediation, where the sorption properties of microbial films play a major role in self-purification. In a biofilm, the microor- ganisms are embedded in a polymeric matrix. This slime layer provides protection to the bacteria and vivid microbial communities can develop. The first generation of mathematical models for biofilms was based on the as- sumption that biofilms develop in homogeneous layers and not much attention was brought to the actual biofilm structure. These models serve well for the purpose of engineering applications on the macro-scale, i.e. on the reactor level [15]. However, they cannot be used to explain the sometimes highly irregular shape of microbial communities and the behavior of biofilms on the meso-scale, i.e. the biofilm itself. 2000 Mathematics Subject Classification. Primary: 35K65; Secondary: 92D25. Key words and phrases. Nonlinear diffusion, degeneracy, global attractor, biofilm. 509