Analysis of chemotactic cross-diffusion in complex frameworks
Final Report Abstract
In the broad active field of research related to chemotactic migration, this project aimed at providing methods of mathematical analysis potentially capable of contributing to a deeper understanding of theoretical aspects thereof. In accordance with recent advances in the biomathematical modeling literature, the objects under investigation have been various classes of partial differential equations which at their core account for taxis-type movement by containing certain cross-diffusion terms as their most characteristic ingredient. The main results identify conditions on the respective model components and parameters which are sufficient either to ensure dominance of taxis-driven destabilization, or to guarantee that such cross-diffusive effects are essentially overbalanced by relaxing mechanisms such as random diffusion or natural saturation effects. This was achieved by analyzing the resulting evolution equations firstly with regard to questions from local and global existence theories, and secondly with respect to aspects of qualitative solution behavior, either in the large time limit or near times and places of possible explosions. Particular contexts in which some progress could thereby be achieved range from the renowned prototypical Keller-Segel model for tactic migration, over chemotaxis systems involving various types of saturation effects in the respective migration mechanisms, possibly moreover accounting for cell proliferation and death, to yet more complex models for chemotactically moving populations, for instance when interacting with liquid environments.
Publications
- A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up. Communications in Partial Differential Equations 42, 436-473 (2017)
Bellomo, N., Winkler , M.
(See online at https://doi.org/10.1080/03605302.2016.1277237) - Finite-time blow-up in a degenerate chemotaxis system with flux limitation. Transactions of the American Mathematical Society B 4, 31-67 (2017)
Bellomo, N., Winkler , M.
(See online at https://doi.org/10.1090/btran/17) - A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization. Journal of Functional Analysis 276, 1339-1401 (2019)
Winkler , M.
(See online at https://doi.org/10.1016/j.jfa.2018.12.009) - Blow-up Profiles for the Parabolic-Elliptic Keller-Segel System in Dimensions n ≥ 3. Communications in Mathematical Physics
Souplet, P H ., Winkler , M.
(See online at https://doi.org/10.1007/s00220-018-3238-1) - Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation. Annales de l’Institut Henri Poincaré – Analyse non linéaire
Winkler , M.
(See online at https://doi.org/10.1016/j.anihpc.2019.02.004) - How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases. Mathematische Annalen
Winkler , M.
(See online at https://doi.org/10.1007/s00208-018-1722-8)
