# zbMATH — the first resource for mathematics

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
##### Keywords:
chemotaxis; global existence; boundedness; blow-up
Full Text:
##### References:
 [1] Hillen, T.; Painter, K.J., A user’s guide to PDE models for chemotaxis, J. math. biol., 58, 183-217, (2009) · Zbl 1161.92003 [2] Keller, E.F.; Segel, L.A., Initiation of slime mold aggregation viewed as an instability, J. theoret. biol., 26, 399-415, (1970) · Zbl 1170.92306 [3] Osaki, K.; Yagi, A., Finite dimensional attractor for one-dimensional keller – segel equations, Funkcialaj ekvac., 44, 441-469, (2001) · Zbl 1145.37337 [4] Nagai, T.; Senba, T.; Yoshida, K., Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. ekvac., 40, 411-433, (1997) · Zbl 0901.35104 [5] Gajewski, H.; Zacharias, K., Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. nachr., 195, 77-114, (1998) · Zbl 0918.35064 [6] Horstmann, D.; Wang, G., Blow-up in a chemotaxis model without symmetry assumptions, European. J. appl. math., 12, 159-177, (2001) · Zbl 1017.92006 [7] Herrero, M.A.; Velázquez, J.J.L., A blow-up mechanism for a chemotaxis model, Ann. sc. norm. super., 24, 633-683, (1997) · Zbl 0904.35037 [8] Jäger, W.; Luckhaus, S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. amer. math. soc., 329, 819-824, (1992) · Zbl 0746.35002 [9] Cieślak, T.; Winkler, M., Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21, 1057-1076, (2008) · Zbl 1136.92006 [10] Kowalczyk, R., Preventing blow-up in a chemotaxis model, J. math. anal. appl., 305, 566-588, (2005) · Zbl 1065.35063 [11] Calvez, V.; Carrillo, J.A., Volume effects in the keller – segel model: energy estimates preventing blow-up, J. math. pures appl., 86, 155-175, (2006) · Zbl 1116.35057 [12] Sugiyama, Y.; Kunii, H., Global existence and decay properties for a degenerate keller – segel model with a power factor in drift term, J. differential equations, 227, 333-364, (2006) · Zbl 1102.35046 [13] Luckhaus, S.; Sugiyama, Y., Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases, Indiana univ. math. J., 56, 1279-1297, (2007) · Zbl 1118.35006 [14] Painter, K.J.; Hillen, T., Volume-filling and quorum-sensing in models for chemosensitive movement, Can. appl. math. Q., 10, 501-543, (2002) · Zbl 1057.92013 [15] Hillen, T.; Painter, K.J., Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. appl. math., 26, 280-301, (2001) · Zbl 0998.92006 [16] Cieślak, T.; Morales-Rodrigo, C., Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect. existence and uniqueness of global-in-time solutions, Topol. methods nonlinear anal., 29, 2, 361-381, (2007) · Zbl 1128.92005 [17] Ladyženskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasi-linear equations of parabolic type, (1968), AMS Providence [18] Friedman, A., Partial differential equations, (1969), Holt, Rinehart & Winston New York [19] Alikakos, N.D., An application of the invariance principle to reaction-diffusion equations, J. differential equations, 33, 201-225, (1979) · Zbl 0386.34046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.