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 $\left\{ \begin{align} & {{u}_{t}}=\Delta u-\chi \nabla \cdot \left( u\nabla v \right)+\kappa u-\mu {{u}^{2}},\ \ \ \ \ \ \ x\in \mathit{\Omega },t>0, \\ & {{v}_{t}}=\Delta v-uv,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathit{\Omega },t>0, \\ \end{align} \right.$ 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 $μ$ and prove that for any $μ>0$ there exists a weak solution.Moreover, in the case of $κ>0$ convergence to the constant equilibrium $(\frac{κ}{μ },0)$ is shown.