Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains

This paper deals with the quasilinear fully parabolic Keller-Segel system {u(t) = del . (D(u)del u) - del . (S(u)del v), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-N with smooth boundary, N is an element of N. The diffusivity D(u) is assumed to satisfy some further technical conditions such as algebraic growth and D(0) >= 0, which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that S(u)/D(u) <= K(u + epsilon)(alpha) for u > 0 with alpha < 2/N, K > 0 and epsilon >= 0. When D(0) > 0, this paper represents an improvement of Tao and Winkler [17], because the domain does not necessarily need to be convex in this paper. In the case Omega = R-N and D(0) >= 0, uniform-in-time boundedness is an open problem left in a previous paper [7]. This paper also gives an answer to it in bounded domains. (C) 2014 Elsevier Inc. All rights reserved.