Analysis of a chemotaxis model with indirect signal absorption

We consider the chemotaxis model {u(t) = Delta u - del center dot (u del v), v(t) = Delta v - uw, w(t) = -delta w + u in smooth, bounded domains Omega subset of R-n, n is an element of N, where delta > 0 is a given parameter. If either n <= 2 or vertical bar vertical bar v(0)vertical bar vertical bar L-infinity (Omega) <= 1/3n we show the existence of a unique global classical solution (u, v, w) and convergence of (u(center dot, t), v(center dot, t), w(center dot, t)) towards a spatially constant equilibrium, as t -> infinity. The proof of global existence for the case n <= 2 relies on a bootstrap procedure. As a starting point we derive a functional inequality for a functional being sublinear in u, which appears to be novel in this context. (C) 2019 Elsevier Inc. All rights reserved.