Existence and stability of infinite time blow-up in the Keller-Segel system

Abstract

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; Δu - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v =&\; (-Δ_{\R^2})^{-1} u := \frac 1{2π} \int_{R^2} \, \log \frac 1{|x-z|}\,u(z,t)\, dz, \\ & \qquad\ u(\cdot ,0) = u_0 \geq 0\quad\hbox{in } R^2. \end{aligned} \right. \end{equation} We consider the {\em critical mass case} ${\int }_{R^2}{u}_0(x)dx=8\pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^{\ast }$ with mass $8\pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^{\ast }$ the solution $u(x,t)$ of \equ{ks0} is globally defined and blows-up in infinite time. As $t\to +\infty$ it has the approximate profile $$u(x,t)\approx \frac{1}{{\lambda }^2}\mathrm{ch}U\left(\frac{x-\xi (t)}{\lambda (t)}\right),\mathrm{ch}U(y)=\frac{8}{(1+|y{|}^2{)}^2},$$ where $\lambda (t)\approx \frac{c}{\sqrt{\log t}},\xi (t)\to q$ for some $c>0$ and $q\in {\mathbb{R}}^2$. This result answers affirmatively the nonradial stability conjecture raised in \cite{g}.

95 pages; final version; comments are welcome

References in corpus (3)

• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Metastability of the proximal point algorithm with multi-parameters
• Singularity formation for the two-dimensional harmonic map flow into $S^2$

Cited by in corpus (1)

• Long-time dynamics of classical Patlak-Keller-Segel equation