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Upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak-Keller-Segel model of chemotaxis

Patrick Maheux

TL;DR

This work derives explicit upper bounds for the blow-up time $T^*$ of the 2-D parabolic-elliptic Patlak–Keller–Segel model under minimal initial-data assumptions ($n_0\ge0$, integrable, mass $M>8\pi$). The authors introduce Gaussian-weighted moments $H_{z,n_0}(s)$ and prove $H_{z,n_0}(T^*)\le \dfrac{2M^2}{3M-8\pi}$, yielding a general bound $T^*(n_0)\le T_c^*(n_0)=\inf_z H_{z,n_0}^{-1}(\dfrac{2M^2}{3M-8\pi})$, with translation-invariance and sharpness considerations. For radially symmetric, non-increasing initial data, a Laplace-transform representation gives an explicit formula $T_c^*(n_0)=\dfrac{1}{4(\mathcal{L}f)^{-1}(L(M)/\pi)}$, tying the bound to the radial profile via $f$ and the mass via $L(M)=\dfrac{2M^2}{3M-8\pi}$. The paper also establishes a lower bound on $T_c^*(n_0)$ through heat-kernel contraction estimates and provides concrete computations for Gaussian and disk initial data, illustrating how the bound depends on both the mass and the initial data’s shape. Overall, the results illuminate how the blow-up time is controlled by both mass and geometry, and they align with the known dichotomy $T^*=\infty$ for $M\le8\pi$ (under additional moment/entropy assumptions) and finite blow-up for $M>8\pi$.

Abstract

In this paper, we obtain upper bounds for the critical time $T^*$ of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the mass is assumed. The result is expressed not only in terms of the supercritical mass $M> 8π$, but also in terms of the {\it shape} of the initial data.

Upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak-Keller-Segel model of chemotaxis

TL;DR

This work derives explicit upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak–Keller–Segel model under minimal initial-data assumptions (, integrable, mass ). The authors introduce Gaussian-weighted moments and prove , yielding a general bound , with translation-invariance and sharpness considerations. For radially symmetric, non-increasing initial data, a Laplace-transform representation gives an explicit formula , tying the bound to the radial profile via and the mass via . The paper also establishes a lower bound on through heat-kernel contraction estimates and provides concrete computations for Gaussian and disk initial data, illustrating how the bound depends on both the mass and the initial data’s shape. Overall, the results illuminate how the blow-up time is controlled by both mass and geometry, and they align with the known dichotomy for (under additional moment/entropy assumptions) and finite blow-up for .

Abstract

In this paper, we obtain upper bounds for the critical time of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the mass is assumed. The result is expressed not only in terms of the supercritical mass , but also in terms of the {\it shape} of the initial data.
Paper Structure (7 sections, 10 theorems, 224 equations)

This paper contains 7 sections, 10 theorems, 224 equations.

Table of Contents

  1. General upper bound on $T^*$ for the (PKS) system
  2. Estimates on the critical time bound $T^*_c(n_0)$
  3. Lower bounds on $T^*_c(n_0)$
  4. Examples of initial data
  5. Gaussian initial data $n_0$
  6. Characteristic function of a disk
  7. More examples of initial data

Key Result

Theorem 1.1

Dongyi Wei Assume that $0\leq n_0\in L^1(\hbox{R}^2)$ and $M=\vert\vert n_0\vert\vert_1$. Let $T^*$ be the maximal existence time of the mild solution. Then $T^*=+\infty$ if and only if $M\leq 8\pi$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 33 more