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

Abstract

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