Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-similar Dynamics in the Critical $p$-Laplacian Patlak-Keller-Segel Model: Shrinking Blow-up and Expanding Propagation

Chunhua Jin, Fengqing Zhang

Abstract

In this paper, we study the following Patlak-Keller-Segel model with $p$-Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &ρ_t=\nabla \cdot \left( \left| \nabla ρ\right|^{p-2}\nabla ρ\right) -χ\nabla \cdot \left( ρ\nabla c \right), &0=\varDelta c+ρ^m, \end{aligned}\right. \end{align*} and the exponent $m>0$ is chosen as $$ m = \frac{(p-2)N + p}{N}. $$ This relation ensures the scale invariance of the system and is conjectured to be the critical exponent that separates global boundedness from finite-time blow-up. We prove that, at the critical threshold $m=\frac{(p-2)N + p}{N}$, the system indeed admits finite-time blow-up solutions. More precisely, in the slow diffusion regime $p>2$, there exist backward self-similar blow-up solutions that are radially decreasing, compactly supported, and concentrate into a Dirac $δ$-measure at the blow-up time $T$; and their supports shrink toward the origin at the rate $(T-t)^{\frac1{mN}}$. For the fast diffusion case $1<p\le 2$, we show that there are no backward self-similar blow-up solutions with finite-mass. Additionally, we also explore forward self-similar solutions in both the slow diffusion and fast diffusion cases. These solutions also carry finite mass and exhibit a Dirac $δ$-singularity at the initial moment. Specifically, in the slow diffusion case, the support expands at the rate $t^{\frac1{mN}}$, whereas in the fast diffusion case, the solution becomes strictly positive for all positive times. Our work provides the first blow up analysis for the $p$-Laplacian Keller-Segel system when $p\ne 2$, and it confirms that the exponent $m$ given above is indeed the sharp threshold between global existence and finite time singularity formation.

Self-similar Dynamics in the Critical $p$-Laplacian Patlak-Keller-Segel Model: Shrinking Blow-up and Expanding Propagation

Abstract

In this paper, we study the following Patlak-Keller-Segel model with -Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &ρ_t=\nabla \cdot \left( \left| \nabla ρ\right|^{p-2}\nabla ρ\right) -χ\nabla \cdot \left( ρ\nabla c \right), &0=\varDelta c+ρ^m, \end{aligned}\right. \end{align*} and the exponent is chosen as This relation ensures the scale invariance of the system and is conjectured to be the critical exponent that separates global boundedness from finite-time blow-up. We prove that, at the critical threshold , the system indeed admits finite-time blow-up solutions. More precisely, in the slow diffusion regime , there exist backward self-similar blow-up solutions that are radially decreasing, compactly supported, and concentrate into a Dirac -measure at the blow-up time ; and their supports shrink toward the origin at the rate . For the fast diffusion case , we show that there are no backward self-similar blow-up solutions with finite-mass. Additionally, we also explore forward self-similar solutions in both the slow diffusion and fast diffusion cases. These solutions also carry finite mass and exhibit a Dirac -singularity at the initial moment. Specifically, in the slow diffusion case, the support expands at the rate , whereas in the fast diffusion case, the solution becomes strictly positive for all positive times. Our work provides the first blow up analysis for the -Laplacian Keller-Segel system when , and it confirms that the exponent given above is indeed the sharp threshold between global existence and finite time singularity formation.
Paper Structure (7 sections, 4 theorems, 206 equations, 5 figures)

This paper contains 7 sections, 4 theorems, 206 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and main results: Backward Self-Similar Blow-up Solutions
  3. Analysis of Backward Self-Similar Blow-up Solutions
  4. The slow diffusion case $p>2$
  5. The fast diffusion case $1<p\leq2$
  6. Forward self-similar singular solution
  7. A brief discussion on the sign structure of the chemical concentration in parabolic-elliptic Keller-Segel model

Key Result

Proposition 2.1

Assume that $N >1$, $p > 2$, and $q > p - 1$. Then the following holds:

Figures (5)

  • Figure 1: Various oscillating behaviours of $u(\cdot , r)$ with $N=1$.
  • Figure 2: Various oscillating behaviours of $u(\cdot , r)$ with $p\geq N>1$.
  • Figure 3: Various oscillating behaviours of $u(\cdot , r)$ with $p\geq N>1$.
  • Figure 4: Behaviour of $u(r, a )$ for $a<a_1$, $a=a_c$ and $a>a_2$ with $p\geq N>1$.
  • Figure 5: $\chi=1$, $N = 4$, $m = \frac{1}{2}$, $p=2$, $u_*= 2\ln 2$. Various oscillating behaviours of $u( \cdot , r)$.

Theorems & Definitions (16)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 6 more