Chemotaxis-consumption interaction: Solvability and asymptotics in general high-dimensional domains

The basic chemotaxis-consumption model $$u_t = \Delta u - \nabla \cdot(u\nabla v),\qquad\qquad v_t = \Delta v - uv$$ is considered in general, possibly non-convex bounded domains of arbitrary spatial dimension. Global existence of weak solutions is shown, along with eventual smoothness of solutions and their stabilization in the large time limit.