Global existence, boundedness and stabilization in a high-dimensional   chemotaxis system with consumption
  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. 
Johannes Lankeit, Yulan Wang
math.AP