Manuel del Pino

University of Bath

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

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

\[\begin{split}\left\{ \begin{aligned} u_t &= Δu - ∇ · (u∇ · v) \qquad \text{ in } \mathbb R^2 × (0,∞), \\ v &= (-Δ)^{-1} u := \frac 1{2π} ∫_{\mathbb R^2} \log \frac 1{|x-z|} u(z,t)\, dz \end{aligned} \right.\end{split}\]

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 \(S^2\),

\[u_t = Δ u + |∇u|^2u \quad \text{ in } \mathbb R^2 × (0,T)\]