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

The basic chemotaxis-consumption model $$u_t=\mathrm{\Delta }u-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v),v_t=\mathrm{\Delta }v-uv$$ 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.