Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source

We consider a quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type, [GRAPHICS] (0.1) with nonnegative initial data under Neumann boundary condition in a smooth bounded domain Omega subset of R-n, n >= 1. Here, phi and psi are supposed to be smooth positive functions satisfying c(1)s(p) <= phi and c(1)s(p) <= psi (s) <= c(2)s(q) when s so with some s(0) > 1, and we assume that g is smooth on [0, co) fulfilling g(0) >= 0 and g(s) <= as - mu s(2) for all s > 0 with constants a >= 0 and mu > 0. Within this framework, it is proved that whenever q < 1, for. any sufficiently smooth initial data there exists a unique classical solution which is global in time and bounded. Our result is independent of p. (C) 2013 Elsevier Inc. All rights reserved.