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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=Δu-χ\nabla\cdot (u\nabla v)+κu-μu^2,\\ v_t=Δv-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 $μ$ and prove that for any $μ>0$ there exists a weak solution. Moreover, in the case of $κ>0$ convergence to the constant equilibrium $(\fracκμ,0)$ is shown.