transactions of the american mathematical society Volume 329, Number 2, February 1992 ON EXPLOSIONS OF SOLUTIONS TO A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS MODELLING CHEMOTAXIS W. JÄGER AND S. LUCKHAUS Abstract. A system of partial differential equations modelling chemotactic ag- gregation is analysed (Keller-Segel model). Conditions on the system of param- eters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria con- centration in finite time is shown for a class of initial values. 1. Introduction The aggregation of organisms sensitive to a gradient of a chemical substance has been of great interest to biologists and mathematicians, trying to model and to simulate the observed pattern formation. The most familiar example of a species showing chemotactic movement is dictyostelium discoideum [G] Model equations were set up and analysed e.g. by Keller and Segel [K-S], W. Alt [Al, A2], and R. Schaaf [S]. The following model was introduced by Keller and Segel to describe the dynamics of a population (concentration u) moving in a domain Í2 and following a gradient of a chemotactic agens (concentration v) produced by the population itself, dtu = Au-xV(uVv) inQ, dtv = yAv - pv + ßu, u(0,-) = u0, v(0,-) = v0, Uo,v0>0, dvu(t,-) = dvv(t,-) = 0 ondQ. X, y, p, ß are positive constants. Due to the experimental facts the diffusion coefficient of the substance v is assumed to be large, of order \ , e small, and ß = ya where a and p are of order 1. From equation (1) we obtain ü(t)=üö, -(dt - p)v = aü = áü~o~, where w denotes (l/\Cl\) Jawdx. Therefore, if we consider v := v -v we get i(d. - p)v = Av - a(u - uö). Hence, for small e we may consider the system (2) dtu = Au-x^iuVv), 0 = Av -aiu-uf). Received by the editors December 7, 1989. 1980 Mathematics Subject Classification ( 1985 Revision).Primary 35B05, 35B30, 35B35, 35B65; Secondary 92A15, 92A17. This work has been supported by the Deutsche Forschungsgemeinschaft (SFB: Stochastísche Mathematische Modelle). ©1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page 819 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use