Long term spatial homogeneity for a chemotaxis model with local sensing and consumption

Global weak solutions to a chemotaxis model with local sensing and consumption are shown to converge to spatially homogeneous steady states in the large time limit, when the motility is assumed to be positive and $C^1$-smooth on $[0,\mathrm{\infty })$. The result is valid in arbitrary space dimension $n\ge 1$ and extends a previous result which only deals with space dimensions $n\in \{1,2,3\}$.