# Manuel del Pino

## University of Bath

### Singularities for the Keller–Segel system in \(\mathbb R^2\)

The Keller–Segel system in \(\mathbb R^2\).

for \(u > 0\), is the classical diffusion model for *chemotaxis*, the motion of a
population of bacteria driven by standard diffusion and a nonlocal drift given
by the gradient of a chemoatractant, a chemical the bacteria produce. It is well
known that mass \(M = ∫_{\mathbb R^2} u(·,t)\) is constant along this flow and
that solutions blows-up in finite time or approach zero as \(t → +∞\) according to
\(M > 8π\) or \(M < 8π\). The critical mass case \(M = 8π\) is more delicate.
Infinite-time blow-up or stabilization around a steady state depends on the
second moment of the solution. We construct a solution globally defined in time
that blows-up as \(t → +∞\) with precise asymptotic profile, establishing stability
of the phenomenon. The method applies to construct solutions that blow up in
finite time simultaneously at several given points in the plane. We discuss the
parallel of this phenomenon in the 2-dimensional harmonic map flow into the
sphere \(S^2\),