Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect

We consider the elliptic-parabolic PDE system {u(t) = del . (phi(u)del u) - del . (psi(u)del v), x is an element of Omega, t > 0, 0 = del v . M + u, x is an element of Omega, t > 0, with nonnegative initial data u(0) having mean value M, under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-n. The nonlinearities phi and psi are supposed to generalize the prototypes phi(u) = (u + 1)(-p), psi(u) = u(u + 1)(q-1) with p >= 0 and q is an element of R. Problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume-filling effect as introduced by Painter and Hillen [K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002) 501-543]. It is proved that if p + q < 2/n then all solutions are global in time and bounded, whereas if p + q > 2/n, q > 0, and Omega is a ball then there exist solutions that become unbounded in finite time. The former result is consistent with the aggregation-inhibiting effect of the volume-filling mechanism; the latter, however, is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume-filling effect in the sense of Painter and Hillen. (C) 2009 Elsevier Ltd. All rights reserved.