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

We consider the elliptic–parabolic PDE system {ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v),x∈Ω,t>0,0=Δv−M+u,x∈Ω,t>0, with nonnegative initial data u0 having mean value M, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn. The nonlinearities ϕ and ψ are supposed to generalize the prototypes ϕ(u)=(u+1)−p,ψ(u)=u(u+1)q−1 with p≥0 and q∈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<2n then all solutions are global in time and bounded, whereas if p+q>2n, q>0, and Ω 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.