Preface: Analysis of cross-diffusion systems
Discrete and Continuous Dynamical Systems. Series S
Directed motion of biological organisms in response to changes of chemical cues in their environment is an ubiquitous mechanism which occurs at all levels of complexity in biological systems, starting from a sub-cellular level through tissue levels up to the interspecies interactions on ecological or social levels. In the 1970s, Keller and Segel proposed their celebrated model for chemotaxis describing movement of unicellular microscopic organisms in response to the gradient of a chemical
... of a chemical signal secreted by the organisms themselves and potentially leading to their eventual aggregation in small spatial regions. The Keller-Segel model and most of its extensions modelling various phenomena related to tumour growth or interspecies interactions in ecology belong to the class of quasilinear parabolic systems with triangular main part but nontrivial off-diagonal entries, also referred to as cross-diffusion systems. During the last few decades many efforts of mathematicians have been focused on the understanding of the singularity formation in finite time in such systems which is related to the process of aggregation and pattern formation. On the other hand, many works were devoted to finding conditions which prevent such finite-time blow-up, thus guaranteeing the existence of global-in-time solutions. Depending on the time scale of biological processes involved some models couple the parabolic equation describing the main component of a system, including the characteristic advective cross-diffusion part, with elliptic equations or even ODEs describing the evolution of the respectively remaining components. Some of the articles included in this volume contain rigorous analysis of new models in mathematical biology describing e.g. tumor growth or virus infection. Other contributions concentrate on new analytical results describing qualitative properties of well-established models.