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Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. (English) Zbl 1183.92012
Summary: We consider the elliptic-parabolic PDE system
\[ \begin{cases} u_t=\nabla\cdot (\phi(u)\nabla u)-\nabla\cdot(\psi(u)\nabla v),\quad & x\in\Omega,\;t>0,\\ 0=\Delta v-M+u, & x\in\Omega,\;t>0,\end{cases} \]
with nonnegative initial data \(u_0\) having mean value \(M\), under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset \mathbb R^n\). The nonlinearities \(\phi\) and \(\psi\) are supposed to generalize the prototypes
\[ \phi(u)=(u+1)^{-p},\quad \psi(u)=u(u+1)^{q-1}, \]
with \(p\geq 0\) and \(q\in\mathbb R\). 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 K. J. Painter and T. Hillen [Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10, No. 4, 501–543 (2002; Zbl 1057.92013)].
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 \(\Omega\) 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.

MSC:
92C17 Cell movement (chemotaxis, etc.)
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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