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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.

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