[Paper Review] Existence and stability of infinite time blow-up in the Keller-Segel system

Juan Dávila, Manuel del Pino+3 more|arXiv (Cornell University)|Nov 27, 2019

Mathematical Biology Tumor Growth50 references4 citations

TL;DR

This paper establishes the existence and nonradial stability of infinite-time blow-up solutions in the 2D Keller-Segel chemotaxis system at critical mass $8\pi$. Using a novel inner-outer gluing scheme, the authors construct a radial initial data $u_0^*$ with mass $8\pi$ and finite second moment, proving that nearby initial data lead to solutions that blow up in infinite time with a self-similar profile $u(x,t) \approx \frac{1}{\lambda(t)^2} U\left(\frac{x - \xi(t)}{\lambda(t)}\right)$, where $\lambda(t) \sim \frac{c}{\sqrt{\log t}}$ and $\xi(t) \to q \in \mathbb{R}^2$, confirming a long-standing conjecture on nonradial stability.

ABSTRACT

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

Motivation & Objective

• To resolve the nonradial stability conjecture for infinite-time blow-up in the critical mass Keller-Segel system.
• To construct a globally defined solution that blows up in infinite time with a precise asymptotic profile.
• To establish the existence of a radial initial condition $u_0^*$ with mass $8\pi$ and finite second moment, such that nearby perturbations yield infinite-time blow-up.
• To prove that the blow-up rate is $\lambda(t) \sim c / \sqrt{\log t}$ and the center $\xi(t)$ converges to a finite point $q \in \mathbb{R}^2$.
• To develop a robust analytical framework for infinite-time blow-up beyond the radial setting.

Proposed method

• Employing an inner-outer gluing scheme to decouple the dynamics near the blow-up point (inner problem) from the global behavior (outer problem).
• Defining the inner variable $y = (x - \xi(t))/\lambda(t)$ to analyze the bubble-like structure of the solution near the blow-up point.
• Using a weighted $C^1$-norm $\|\phi\|_{*\sigma}$ to measure perturbations around the initial data $u_0^*$, ensuring sufficient regularity and decay.
• Constructing supersolutions via barrier functions $\psi_1$, $\psi_2$, and $\tilde{\psi}$ to control error terms in the outer and inner regions.
• Deriving and solving a nonlocal ODE system for the center $\xi(t)$, which governs the motion of the blow-up bubble.
• Applying spectral estimates and energy methods to control the linearized operator $L_o$ in the outer region, ensuring stability under nonradial perturbations.

Experimental results

Research questions

• RQ1 Can infinite-time blow-up occur in the 2D Keller-Segel system for initial data near a critical mass radial profile, even under nonradial perturbations?
• RQ2 What is the precise asymptotic rate of blow-up for the scaling parameter $\lambda(t)$ in the infinite-time regime?
• RQ3 Does the center $\xi(t)$ of the blow-up profile converge to a finite point in $\mathbb{R}^2$ as $t \to \infty$?
• RQ4 Can the stability of the blow-up profile be established in the nonradial class, beyond the radial setting?
• RQ5 What is the role of the second moment in determining the blow-up behavior at critical mass $8\pi$?

Key findings

• There exists a radial initial condition $u_0^*$ with mass $8\pi$ and finite second moment such that all sufficiently close perturbations (in $C^1$-weighted norm) lead to global solutions that blow up in infinite time.
• The blow-up profile satisfies $u(x,t) \approx \frac{1}{\lambda(t)^2} U\left(\frac{x - \xi(t)}{\lambda(t)}\right)$ uniformly on bounded sets, with $U(y) = \frac{8}{(1 + |y|^2)^2}$.
• The scaling parameter satisfies $\lambda(t) = \frac{c}{\sqrt{\log t}} (1 + o(1))$ as $t \to \infty$, confirming the formal rate predicted in earlier works.
• The center $\xi(t)$ converges to a finite limit $q \in \mathbb{R}^2$ as $t \to \infty$, indicating a stable spatial localization of the blow-up.
• The solution is stable under nonradial perturbations, confirming the nonradial stability conjecture raised in [26].
• The construction relies on a new inner-outer gluing scheme that successfully handles the slow blow-up rate and nonlocal dynamics.