The Dichotomy of Global Existence and Blow-up in Chemotaxis Systems
Dott. Rafael Díaz Fuentes
Università di Cagliari
Abstract
This course provides a comprehensive analysis of systems of nonlinear partial differential equations, primarily of parabolic and elliptic type, that arise in the modeling of chemotaxis. The curriculum is structured around the fundamental dichotomy between two possible fates for solutions: global existence and finite-time blow-up. The analysis focuses on the critical thresholds in mass, dimension, and model parameters that determine the outcome of a dynamic competition between stabilizing forces (diffusion, logistic damping) and destabilizing forces (aggregation). Beginning with the classical theory, the course progresses to several advanced topics, including: the mechanisms that enforce global regularity, the long-term behavior of such solutions, and the modern analytical tools required to establish well-posedness in challenging low-regularity settings.
Schedule
The course will be held in April-May 2026, delivered over 8 lectures for a total of 14-16 hours.
Exam
The final assessment consists of one of the two choices: a seminar on a topic building on the content of the course or a written elaborate on selected topics of the course.
References
- Bellomo, N., Belloquid, A., Tao, Y., & Winkler, M. (2015). Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Mathematical Models and Methods in Applied Sciences, 25(9), 1663-1763.
- Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
- Evans, L. C. (2010). Partial Differential Equations (2nd ed.). American Mathematical Society.
- Horstmann, D. (2003). From 1970 until now: the Keller-Segel model in chemotaxis and its consequences. I. Jahresbericht der Deutschen Mathematiker-Vereinigung, 105(3), 103-165.
- Lankeit, J., Winkler, M. (2017). Facing Low Regularity in Chemotaxis Systems. Jahresbericht der Deutschen Mathematiker-Vereinigung, 122, 35-64.
- Winkler, M. (2010). Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. Journal of Differential Equations, 248(12), 2889-2905.