Explosions and solvability in cross-diffusive systems of chemotaxis type

• Applicant: Professor Dr. Johannes Lankeit
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552796953

Chemotaxis systems are model problems for self-organization, teetering on the brink between global solvability and explosions ('blow-up') of solutions as the most striking form of spontaneous emergence of structure. This project aims at expanding the knowledge about this important class of Partial Differential Equations. In particular, it will establish, and more closely investigate, blow-up in a fully parabolic chemotaxis system with source terms of logistic type, and thus in a class of models which has, up to now, remained inaccessible to any known blow-up method. Beyond this, it will address different aspects of emerging structure, especially those already occurring on shorter time-scales and not only visible in the limit of large times. Particularly when, due to the occurrence of blow-up, classical solvability cannot be achieved, or when it is unclear whether it may be expected, weaker solution concepts become relevant. Part of the project will thus be a closer investigation of generalized solution concepts along with their relations with each other and with the possibility of blow-up. Apart from questions of existence and solvability, this includes questions on extensibility after blow-up. Applications will include simple model systems as well as more complex situations, e.g. with multi-stage taxis mechanisms or spatially inhomogeneous coefficients arising in spatial ecology or in an immunological context.

• DFG Programme: Research Grants