On a parabolic–elliptic chemotactic model with coupled boundary conditions

Abstract

This paper deals with a nonlinear system of parabolic–elliptic type with a logistic source term and coupled boundary conditions related to pattern formation. We prove the existence of a unique positive global in time classical solution. We also analyze the associated stationary problem. Moreover it is proved, under the assumption of sufficiently strong logistic damping, that there is only one nonzero homogeneous equilibrium, and all the solutions to the nonstationary problem tend to this steady state for large times.