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 $p$-Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &\rho _t=\nabla \cdot \left( \left| \nabla \rho \right|^{p-2}\nabla \rho \right) -\chi \nabla \cdot \left( \rho \nabla c \right), &0=\varDelta c+\rho ^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 $\delta$-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 $\delta$-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.