Global existence and blow-up phenomena in Keller-Segel-Type systems with tensorial flux

Abstract

In the past decade, there has been much interest in analyzing Keller-Segel
models with tensorial flux. However, it is not yet well understood whether
there are solutions that blow-up in a finite amount of time. This thesis aims
to bridge this gap by developing a comprehensive approach capable of yield-
ing sharp results regarding global existence and blow-up phenomena across
various systems characterized by the interplay between vorticity and one or
more chemotactic signals. Furthermore, significant progress has been made
in resolving numerous open problems pertaining to the existence of solutions
for diverse mathematical models in the realms of mathematical physics and
biology, cf. [82, 87, 92, 93]. Moreover, Significant advancements have been
made in analyzing Keller-Segel models with tensorial flux in both two and
higher dimensions, achieved through the introduction of a novel technique.
This technique showcases the possibility of finite-time blowup solutions in the
Keller-Segel model, even under highly general conditions on the tensorial flux.
On the other hand, cells encounter a diverse array of physical and chemical
signals as they navigate their natural surroundings. However, their response
to the simultaneous presence of multiple cues remains elusive. Particularly,
the impact of topography alongside a chemotactic gradient on cell migratory
behavior remains insufficiently explored. So, it is noteworthy that among the
innovations of this thesis, we also delve into analyzing the conditions that pre-
dict or prevent cell aggregation when obstacles interfere during the process.

Date of Award	Oct 2024
Original language	English
Awarding Institution	University of Nottingham
Supervisor	Behrouz Emamizadeh (Supervisor), Elio Eduardo Espejo Arenas (Supervisor) & Richard Rankin (Supervisor)