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

DOI: http://dx.doi.org/10.12775/TMNA.2007.015

#### 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.

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.

#### Keywords

Chemotaxis equations; global-in-time existence and uniqueness; quasilinear reaction-diffusion systems; prevention of blow-up; Schauder fixed point theorem

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