Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant

This paper deals with positive solutions of{ut=Δu−∇⋅(u∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, under homogeneous Neumann boundary conditions in bounded convex domains Ω⊂R3Ω⊂R3 with smooth boundary.It is shown that for arbitrarily large initial data, this problem admits at least one global weak solution for which there exists T>0T>0 such that (u,v)(u,v) is bounded and smooth in Ω×(T,∞)Ω×(T,∞). Moreover, it is asserted that such solutions approach spatially constant equilibria in the large time limit.