Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect. Existence and uniqueness of global-in-time solutions

Abstract

A system of quasilinear non-uniformly parabolic-elliptic equations
modelling chemotaxis and taking into account the volume filling
effect is studied under no-flux boundary conditions. The proof of
existence and uniqueness of a global-in-time weak solution is given.
First the local solutions are constructed. This is done by the
Schauder fixed point theorem. Uniqueness is proved with the use of
the duality method. A priori estimates are stated either in the case
when the Lyapunov functional is bounded from below or chemotactic
forces are suitably weakened.