On the stability of homogeneous steady states of a chemotaxis system with logistic growth term

We consider a nonlinear PDEs system of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a population “” towards a higher concentration of a chemical “” in a bounded domain.

We consider constant chemotactic sensitivity and an elliptic equation to describe the distribution of the chemical
for a monotone increasing and Lipschitz function .

We study the asymptotic behavior of solutions under the assumption of . As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.