Boundedness enforced by mildly saturated conversion in a chemotaxis-May Nowak model for virus infection

We study the system {u(t) = Delta(u) - del . (u del v) - u - f(u)w + kappa, v(t) = Delta v - v + f(u)w, w(t) = Delta w - w + v, (star) which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain Omega subset of R-n, n is an element of N, for a parameter kappa >= 0 and a given function f is an element of C-1 ([0, infinity)) satisfying f >= 0, f (0) = 0 and f(s) <= K(f)s(alpha) for all s >= 1, some K-f > 0 and alpha is an element of R. We prove that whenever alpha < 2/n, solutions to (star) exist globally and are bounded. The proof mainly relies on smoothing estimates for the Neumann heat semigroup and (in the case alpha > 1) on a functional inequality. Furthermore, we provide some indication why the exponent 2/n could be essentially optimal. (C) 2018 Elsevier Inc. All rights reserved.