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

Abstract

This paper deals with positive solutions of { u t = Δ u − ∇ ⋅ ( u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in bounded convex domains Ω ⊂ 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 Ω × ( T , ∞ ) . Moreover, it is asserted that such solutions approach spatially constant equilibria in the large time limit.