Global boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with logistic source

We consider a quasilinear parabolic parabolic Keller-Segel system with a logistic type source u(t) = del . (phi(u)del u) - del . (psi(u)del v) + g(u), v(t) = Delta v - v + u in a smooth bounded domain Omega subset of R-n, n >= 1, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where phi, psi and g are smooth positive functions satisfying c(1)s(p) <= phi(s) and c(1)s(q) <= psi(s) <= c(2)s(q) for p, q is an element of R s >= s(0) > 1 g(s) <= as - mu s(k) for s > 0, with constants a >= 0, mu, c(1), c(2) > 0, and the extended logistic exponent k > 1 instead of the ordinary k = 2. It is proved that if q < k - 1, or q = k - 1 with mu properly large that mu > mu(0) for some mu(0) > 0, then there exists a classical solution which is global in time and bounded. This shows the exact way of the logistic exponent k > 1 effecting the behavior of solutions. (C) 2015 Elsevier Ltd. All rights reserved.