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

Abstract

We consider a nonlinear PDEs system of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a population “ n ” towards a higher concentration of a chemical “ c ” in a bounded domain Ω . We consider constant chemotactic sensitivity χ and an elliptic equation to describe the distribution of the chemical n t − d n Δ n = − χ div ( n ∇ c ) + μ n ( 1 − n ) , − d c Δ c + c = h ( n ) for a monotone increasing and Lipschitz function h . We study the asymptotic behavior of solutions under the assumption of 2 χ | h ′ | < μ . As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.