Nonlinear Functional Analysis and Variational Methods for Elliptic Systems Arising in Chemotaxis Models of Cell Movement

This dissertation develops original results in nonlinear functional analysis and the calculus of variations for elliptic and parabolic systems arising in mathematical models of chemotaxis, the directed movement of cells in response to chemical gradients, with applications to wound healing dynamics and tumour microenvironment modelling of direct relevance to understanding cancer biology in the Nigerian population context. Chemotaxis modelling via Keller-Segel type systems presents fundamental analytical challenges including blow-up singularity formation in finite time, the competition between diffusion-driven spread and chemotactic aggregation, and the mathematical regularity of steady-state spatial pattern forming solutions, all of which require advanced tools from nonlinear functional analysis and modern elliptic regularity theory. The dissertation proves novel global existence theorems for a class of fully parabolic Keller-Segel systems with volume-filling effects in spatial dimensions two and three, using energy method arguments in Orlicz spaces that provide sharper critical mass thresholds than those obtainable by Lp energy estimates. New variational characterisations of chemotaxis steady states as constrained critical points of a free energy functional are established, and Morse theory methods are applied to count the number of distinct positive steady-state solution branches as system parameters vary. The relationship between the cell aggregation blow-up phenomenon and the capacity theory of fractional Sobolev spaces is explored, establishing original necessary conditions on initial cell density for global solutions in terms of the Bessel capacity of the support. Keywords: chemotaxis, Keller-Segel system, nonlinear functional analysis, Orlicz spaces, blow-up theory