This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{ eqnarray*} \begin{ array}{ llc} u_t=\Delta u-\chi\nabla\cdot (u\nabla v)+\kappa u-\mu u^2,\\ v_t=\Delta v-uv, \end{ array} \end{ eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $\mu$ and prove that for any $\mu>0$ there exists a weak solution. Moreover, in the case of $\kappa>0$ convergence to the constant equilibrium $(\frac{ \kappa}{ \mu},0)$ is shown.
Categories: math.AP