Page 67 -

4 Chemotactic Cell Motion and Biological Pattern Formation
60

differential equations exhibits a fascinating feature, under certain conditions on
the initial datum

r(x, t = 0) = r 0 , (4.5)

namely finite time blow-up of solutions. More precisely, two cases have to be
distinguished for n = 2 (two space dimensions!), where we denote the total
initial cell mass by:

M 0 := r 0 (x) dx .

2
These cases are:
Case A: M 0 < 8πD 0/c. Then a global weak solution of (4.1), (4.4), (4.5) exists.
Case B: M 0 > 8πD 0/c.Thensolutions r of (4.1), (4.4), (4.5) blow up in finite
time, global in time solutions do not exist.

Usually, Case Aisreferredtoassubcritical case (massissmall enough such that
finite time blow up can be avoided) and Case B (mass is too big, finite time blow
up occurs) as supercritical. There is no finite time blow up for the one dimen-
sionalclassicalKeller–Segelmodelwhilethreedimensionalsolutionsgenerically
concentrate in finite time. We remark that the mechanism, which inhibits the
global existence of solutions in the supercritical case, is concentration of the
cell density, i.e. r(x, t)tends locallytoaDirac-δ distribution when t approaches
a finite blow-up time T. Beyond blow up time the solutions cannot be extended
without somewhat redefining the problem. The reason for the non-existence of
time-global solutions is that the production of the chemical by the cells generates
an attractive force field, just as for PDE models of gravitational particles. This
is totally different from the situation where the cells (presumably) destroy the
chemical (which corresponds to changing the sign of the chemotactic sensitiv-
ity c in (4.1)), analogously to the repulsive Coulomb force acting on the charged
particles in semiconductors, modeled by the semiconductor drift-diffusion sys-
tem. To better understand the mechanism of an attractive resp. repulsive force,
choose a number q> 1, multiply the Fokker–Planck equation (4.1) by qr q−1 and
n
integrate over R . Then, after integration by parts and using (4.4) we obtain,
assuming that c is constant:
d q 2 q−2 q+1
r(x, t) dx = − q(q −1)D 0 |∇r| r dx +(q −1)c r dx .
dt

Thus, the right hand side is nonpositive if c is nonpositive (repulsive case) and
q
consequently the L -norm of the position density r is uniformly bounded for
t> 0. Clearly, this excludes a concentration in the density r. Note that this
argument fails in the case of an attractive force c> 0!