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Existence of finite time blow-up in Keller-Segel system

Federico Buseghin, Juan Davila, Manuel del Pino, Monica Musso

Abstract

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =Δu - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-Δ_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2π} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8π<m\le 8π+\varepsilon$ and any $k$ given points $q_{1},...,q_{k}$ in $\mathbb{R}^{2}$ there is an initial data $u_0^*$ of $(\star)$ for which the solution $u(x,t)$ blows-up in finite time as $t\to T$ with the approximate profile $$u(x,t)=\sum_{j=1}^{k}\frac{1}{λ_{j}^{2}(t)}U\left(\frac{x-ξ_{j}(t)}{λ_{j}(t)}\right)(1+o(1)), U(y)=\frac{8}{(1+|y|^{2})^{2}},$$ with $λ_{j}(t) \approx 2e^{-\frac{γ+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}} $ where $γ=0.57721...$ is the Euler-Mascheroni constant, $ξ_{j}(t)\to q_{j}\in \mathbb{R}^{2}$ and such that $\int_{\mathbb{R}^2}u(x,t)dx=km.$ This construction generalizes the existence result of the stable blow-up dynamics recently proved in \cite{CGMN1,CGMN2}.

Existence of finite time blow-up in Keller-Segel system

Abstract

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system We show that there exists such that for any satisfying and any given points in there is an initial data of for which the solution blows-up in finite time as with the approximate profile with where is the Euler-Mascheroni constant, and such that This construction generalizes the existence result of the stable blow-up dynamics recently proved in \cite{CGMN1,CGMN2}.
Paper Structure (10 sections, 18 theorems, 308 equations)

This paper contains 10 sections, 18 theorems, 308 equations.

Table of Contents

  1. Introduction
  2. The approximations $u_{0}$ and $u_{1}$
  3. The first error of approximation
  4. The inner-outer gluing system
  5. Choice of $\lambda_{0}$, $\alpha_{0}$ and $\xi_{0}$
  6. A further improvement of the approximation
  7. Reformulation of the system
  8. Splitting the inner solution $\phi$
  9. Mass, second moment and center of mass
  10. Proof of Theorem \ref{['theorem1']}

Key Result

Theorem 1.1

There exists $\varepsilon>0$ such that for any $m$ satisfying and any $k$ given points $q_{1},...,q_{k}$ in $\mathbb{R}^{2}$ there is an initial data $u_0^{\star}$ of (ks0) for which the corresponding solution $u(x,t)$ has the form uniformly on bounded sets of $\mathbb{R}^{2}$, with $\lambda_{j}(t) \approx 2e^{-\frac{\gamma+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}}$ where $\gamma=0.57721.

Theorems & Definitions (34)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 24 more