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 $\{ {align} & {{u}ₜ}=Δ u-χ ∇ · ( u∇ v )+κ u-μ {{u}²},\ \ \ \ \ \ \ x∈ Ω ,t>0, \\ & {{v}ₜ}=Δ v-uv,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x∈ Ω ,t>0, \\ {align} .$ 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 $(κ/μ ,0)$ is shown.