Long-term behaviour in a parabolic-elliptic chemotaxis-consumption model

Abstract

Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= Δn - \nabla \cdot (n \nabla c), \\ 0 &= Δc - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions \[ \partial_ν c = (γ-c) g \] for $c$ (with $\gamma >0$ and $C^{1+\beta }(\partial \Omega )\ni g>0$ for some $\beta \in (0,1)$) are established in bounded domains $\Omega \subset {\mathbb{R}}^N$, $N\ge 1$. Under a smallness condition on $\gamma$, moreover, we show convergence to the stationary solution.