Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source

We study global solutions of a class of chemotaxis systems generalizing the prototype {u(t) = del center dot ((u + 1)(m-1)del u) - chi del center dot (u(u + 1)(q-1)del nu) + au - bu(r), x is an element of Omega, t > 0, 0 = Delta nu - nu + u, x is an element of Omega, t > 0, in a bounded domain Omega subset of R-N (N >= 1) with smooth boundary, with parameters m >= 1, r > 1, a >= 0, b, q, chi > 0. It is shown that when q + 1 < max{r, m + 2/N}, or b > b(0) := N[r - m] - 2/(r - m)N + 2(r - 2) chi if q + 1 = r, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results improve the results of Wang et al. (2014) [37] and Cao and Zheng (2014) [6]. (C) 2015 Elsevier Inc. All rights reserved.