Summary: The parabolic-parabolic Keller-Segel system for chemotaxis phenomena,
\[ \begin{cases} u_t=\nabla\cdot (\varphi(u)\nabla u)-\nabla\cdot(\psi(u)\nabla v), &x\in\Omega,\;t>0,\\ v_t=\Delta v-v+u, &x\in\Omega,\;t>0,\end{cases} \]
is considered under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb R^n\) with \(n\geq 2\).
It is proved that if \(\psi(u)/\varphi(u)\) grows faster than \(u^{2/n}\) as \(u\to\infty\) and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass \(\int_\Omega u(x,t)\,dx\) may attain arbitrarily small positive values.
In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of K. J. Painter and T. Hillen [Can. Appl. Math. Q. 10, No. 4, 501–543 (2002; Zbl 1057.92013)], the results indicate how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse.