Problem arising from Mathematical Biology solved
Paper The full Keller-Segel model is well-posed on nonsmooth domains by Dirk Horstmann (University of Cologne), Hannes Meinlschmidt (Johann Radon Institute Linz) and Joachim Rehberg (WIAS) published in Nonlinearity.
The Keller-Segel Model describes the interaction of bacteria and their surrounding agents. It was established by Keller and Segel in 1970 and is probably the most analysed model in Mathematical Biology. Their work was citated severel hundreds of times and it became an initial point of various analytical and numerical studies. The model is of high relevance for the description of clinical phenomena. The (full) Keller-Segel Model consists of four strongly coupled differential equations. While each of these equations is parabolic itself, the same does not hold necessarily for the whole system. The equations are applied for a prescribed spatial or planar area (like a Petri dish) containing the bacteria culture. In most cases this system is considered to be isolated from the outside world.
For areas with smooth geometries (e.g. round or ellipsoid) it has been known for quite a long time that for given initial values for bacteria density and chemical agents, a unique solution exists at least for a short time interval. However, in practice the considered areas are typically nonsmooth. In particular at geometrically singular areas like corners or edges, exceptional effects like extra high concentrations are supposed to occur. Now for the first time the three authors could prove the existence of a solution for the Keller-Segel Model also for a huge class of non-smooth geometries like arbitrary two- and three-dimensional polyhedral areas.
The paper was published in Nonlinearity:
D. Horstmann, H. Meinlschmidt and J. Rehberg: The full Keller-Segel model is well-posed on nonsmooth domains. Nonlinearity (2018).