Does a ‘volume-filling effect’ always prevent chemotactic collapse? 论文

摘要

The parabolic–parabolic Keller–Segel system for chemotaxis phenomena, is considered under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ℝn with n⩾2. It is proved that if ψ(u)/ϕ(u) grows faster than u2/n as u→∞ and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass ∫Ωu(x, t)dx may attain arbitrarily small positive values. In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4):501–543), the results indicate how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Copyright © 2009 John Wiley & Sons, Ltd.