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Analysis of chemotactic cross-diffusion in complex frameworks

Subject Area Mathematics
Term from 2016 to 2020
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 288366228
 
Final Report Year 2019

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.

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