Asymptotics of a chemotaxis-consumption-growth model with nonzero Dirichlet conditions

This paper concerns the asymptotics of certain parabolic-elliptic chemotaxis-consumption systems with logistic growth and constant concentration of chemoattractant on the boundary. First we prove that in two dimensional bounded domains there exists a unique global classical solution which is uniformly bounded in time, and then we show that if the concentration of chemoattractant on the boundary is sufficiently low then the solution converges to the positive steady state as time goes to infinity.