In this article, we give a brief overview of our recent work on continuum mechanical modelling and simulation of microbial films. This comprises some classical tasks of applied mathematics such as computational fluid dynamics, analysis of partial differential equations, and mathematical biology.

Anonlinear density-dependent system of diffusion-reaction equations describing the spatial spreading of biomass during the development of microbial films is analysed. It comprises two non-standard diffusion effects, degeneracy as in the porous medium equation and fast diffusion. The existence of aunique bounded so lution and aglobal attractor is proved in dependence of the boundary conditions. This is achieved by studying an auxiliary approximating sequence of systems of nondegenerate evolution equations and the construction of aLipschitz continuous semigroup by passing to the limit in the approximation parameter. Numerical examples are given that illustrate the main result of this paper. INTRODUCTION. Biofilms play avery important role in many scientific and technological areas. Consequently, they are studied in many disciplines and biofilm research is atruly interdisciplinary research topic. Biofilms are the most succesful life form on earth growing virtually everywhere, where nutrients are available to feed bacteria. In fact, most bacteria live in biofilm colonies and only asmall minority appears as suspended planktonic organisms. Biofouling, biocorrosion, and bacterial infections are harmful impacts of biofilms. On the other hand, benefical properties of biofilms are used in enviromental engineering for wastewater treatment, groundwater protection, and soil remediation, where the sorption properties of microbial films play akey role in self-purification. The microorganisms in abiofilm are embedded in apolymeric 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 assumption that biofilms develop in flat homogeneous layers and not much attention was brought to the actual biofilms structure. These models serve well for the purpose of engineering applications on the macr0-scale, $i.e$ . on the reactor level. However, they cannot be used to explain the sometimes highly irregular shape of microbial communities and the behavior of biofilms on the mes0-scale, $i.e$ . the biofilm itself. Since this first generation of biofilm models based on the seminal work [WG86] explicitly takes advantage of the one-dimensionality of the model setup (the biofilm can only grow perpendicularly to the substratum), ageneralization of this approach to the spatially heterogeneous case is not possible. Therefore new model concepts became necessary. The big challenge in biofilm modelling is to describe the spatial spreading mechanism for biomass. The a priori postulations for most spatial biofilms, derived ffom experimental evidence, are: Typeset by $A\mathcal{M}\theta \mathfrak{M}$ 数理解析研究所講究録 1258巻 2002年 49-71