Herrero, Miguel A.; Velázquez, Juan J. L. Singularity patterns in a chemotaxis model. (English) Zbl 0864.35008 Math. Ann. 306, No. 3, 583-623 (1996). Summary: Consider the system \[ u_t=\Delta u-\chi\nabla(u\nabla\nu),\qquad 0=\Delta\nu+(u-1)\quad \text{for }x\in\Omega,\;t>0, \tag{S} \] together with no-flux boundary conditions \({{\partial u}\over{\partial n}}={{\partial\nu}\over{\partial n}}=0\) for \(x\in\partial\Omega\), \(t>0\), and initial values \(u(x,0)= u_0(x)\), where \(u_0(x)\) is, say, a continuous and nonnegative function. Here, \(\Omega\) denotes a smooth and bounded open set in \(\mathbb{R}^2\), and \(\chi>0\). System (S) is a model to describe chemotaxis for a species (whose concentration is represented by \(u(x,t)\) under the action of a chemical (whose concentration is denoted by \(\nu(x,t)\)) which is secreted by the species organisms. We prove that if \(\Omega\) is a ball, there exist radial solutions of that problem, which concentrate into a Dirac mass in finite time. This fact is usually referred to as chemotactic collapse. The manner in which chemotactic collapse develops in our system is analyzed in detail. Cited in 5 ReviewsCited in 152 Documents MSC: 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) 35B40 Asymptotic behavior of solutions to PDEs Keywords:blow-up; radial solutions; chemotactic collapse × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] [A1] W. Alt: Orientation of cells migrating in a chemotatic gradient. Lectures Notes in Biomath. vol.38, Springer-Verlag (1980), 353-366. [2] [A2] W. Alt: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol.9 (1980), 147-177. · Zbl 0434.92001 · doi:10.1007/BF00275919 [3] [AV] SB. Angenent, JJ.L. Velázquez: Degenerate neckpinches in mean curvature flow. To appear. [4] [BGG] E. Bombieri, E. De Giorgi, E. Giusti: Minimal cones and the Bernstein problem. Inventiones Math.7 (1969), 243-268. · Zbl 0183.25901 · doi:10.1007/BF01404309 [5] [C] S. Childress: Chemotactic collapse in two dimensions. Lectures Notes in Biomath. vol.55, Springer-Verlag (1984), 61-66. [6] [CP] S. Childress, JK. Percus: Nonlinear aspects of chemotaxis. Math. Biosci.56 (1981), 217-237. · Zbl 0481.92010 · doi:10.1016/0025-5564(81)90055-9 [7] [DN] J.I. Diaz, T. Nagai: Symmetrization in a parabolic-elliptic system related to chemotaxis. Adv. Math. Sci. Appl., to appear. · Zbl 0859.35004 [8] [GK] Y. Giga, RV. Kohn: Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure. Appl. Math.38 (1985), 297-319. · Zbl 0585.35051 · doi:10.1002/cpa.3160380304 [9] [HV1] MA. Herrero, JJL. Velazquez: Explosion des solutions d’équations paraboliques semilinéaires supercritiques. C.R. Acad. Sci. Paris t.319, I (1994), 141-145. [10] [HV2] MA. Herrero, J JL. Velazquez: On the melting of ice balls. SIAM J. Math. Anal, to appear. [11] [JL] W. Jäger, S. Luckhaus: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc., vol.329, n. 2 (1992), 819-824. · Zbl 0746.35002 · doi:10.2307/2153966 [12] [KS] EF. Keller, LA. Segel: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol.26 (1970), 399-415. · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5 [13] [N] T. Nagai: Blow-up of radially symmetric solutions to a chemotaxis system. To appear in Adv. Math. Sci. Appl. · Zbl 0843.92007 [14] [Na] V. Nanjundiah: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol.42 (1973), 63-105. · doi:10.1016/0022-5193(73)90149-5 [15] [V] JJL. Velazquez: Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow. Annali Scuola Normale Superiore di Pisa, Serie IV, vol.XXI, Fasc. 4 (1994), 595-628. · Zbl 0926.35023 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.