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 “nn” towards a higher concentration of a chemical “cc” in a bounded domain ΩΩ.We consider constant chemotactic sensitivity χχ and an elliptic equation to describe the distribution of the chemical nt−dnΔn=−χdiv(n∇c)+μn(1−n),−dcΔc+c=h(n) for a monotone increasing and Lipschitz function hh.We study the asymptotic behavior of solutions under the assumption of 2χ|h′|<μ2χ|h′|<μ. As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.