Eventual smoothness and stabilization of global weak solutions to a chemotaxis system with sublinear consumption of chemoattractant

This paper investigates a chemotaxis system with sublinear consumption of chemoattractant$\begin{align*} \begin{cases} u_t = \Delta u-\nabla \cdot(u \nabla v), \\ v_t = \Delta v-u v^m \end{cases} \end{align*} $under homogeneous Neumann boundary conditions in bounded convex domain $ \Omega \subset \mathbb{R}^3 $ with smooth boundary. We prove that if $ m \in (0, 1] $, for arbitrarily large initial data, then the model ($\star$) admits at least one global weak solution for which there exists $ T>0 $ such that $ (u, v) $ is bounded and smooth in $ {\Omega} \times(T, \infty) $. Moreover, it is asserted that such solutions are seen to stabilize toward spatially constant equilibrium in the large time limit.