Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system

TL;DR

This paper investigates the critical case of the one-dimensional quasilinear Smoluchowski-Poisson system, identifying conditions for global existence versus finite-time blow-up based on the diffusion coefficient.

Contribution

It introduces a new change of variables and virial identity to distinguish diffusion classes leading to different solution behaviors.

Findings

All solutions are global for the critical case p=1.

Two classes of diffusion coefficients are identified: one with global solutions, another with blow-up.

A novel analytical approach was developed to analyze solution behavior.

Abstract

It is known that classical solutions to the one-dimensional quasilinear Smoluchowski-Poisson system with nonlinear diffusion $a(u)=(1+u)^{-p}$ may blow up in finite time if $p>1$ and exist globally if $p<1$. The case $p=1$ thus appears to be critical but it turns out that all solutions are global also in that case. Two classes of diffusion coefficients are actually identified in this paper, one for which all solutions to the corresponding quasilinear Smoluchowski-Poisson system are global and the other one leading to finite time blow-up for sufficiently concentrated initial data. The cornerstone of the proof are an alternative formulation of the Smoluchowski-Poisson system which relies on a novel change of variables and a virial identity.