The impact of Robin boundary condition on a chemotaxis-consumption-growth model

We investigate a parabolic-elliptic chemotaxis-consumption-growth system with a Robin boundary condition imposed on the signal. First, we analyse the steady state problem, then we show that the solutions of the system are global and uniformly bounded in time in any space dimension. Next, under smallness assumption on the boundary data, we show that the solutions converge to non-constant steady states as time tends to infinity.