Maximal spaces and threshold values for PDEs modelling chemotaxis

Abstract

In this project we are interested in chemotaxis models (including the famous Keller-Segel system) which describe the dynamic of cell systems such as populations of bacteria, protozoa, and somatic cells. Such dynamic allows aggregation and formation of singularity (in a way depending on the dimension $n\geq2$), which motivates to study the corresponding PDEs in spaces that contain singular elements, especially measures supported on points and curves. In this direction, we intend to obtain new maximal classes for existence of global solutions, analysing the PDEs in Fourier-Besov-Morrey spaces and $Q$-type-spaces. Another issue of interest is the analysis of possible threshold values that determine a dichotomy in the systems with respect to global existence or formation of singularity at finite time. In this part, we intend to use spaces that allow to obtain explicit estimates for those values by means of an approach without using entropy methods and symmetry arguments via rearrangements. We will also investigate self-similar solutions, and existence and asymptotic stability of stationary solutions in the above spaces.