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

Abstract The parabolic–parabolic Keller–Segel system for chemotaxis phenomena,\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}\left\{\begin{array}{ll} u_t=\nabla\cdot(\phi(u)\nabla u)-\nabla\cdot(\psi(u)\nabla v), & x\in\Omega,\, t{>}0\\ v_t=\Delta v-v+u, & x\in\Omega,\, t{>}0\end{array}\right.\end{eqnarray*}\end{document}is considered under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ℝ n with n ⩾2. It is proved that if ψ( u )/ϕ( u ) grows faster than u 2/ 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 ∫ Ω u ( x, t )d x 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.