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 { u(t) = Delta u - del . (u del v), x is an element of Omega, t > 0, v(t) = Delta v - uv, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in bounded convex domains Omega subset of R(3) 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 > 0 such that (u, v) is bounded and smooth in Omega x (T, infinity). Moreover, it is asserted that such solutions approach spatially constant equilibria in the large time limit. (C) 2011 Elsevier Inc. All rights reserved.