Global existence vs. blowup for the one dimensional quasilinear Smoluchowski-Poisson system

TL;DR

This paper demonstrates that in one dimension, the quasilinear Smoluchowski-Poisson system does not have a critical diffusion coefficient separating global existence from blowup, unlike in higher dimensions.

Contribution

It establishes the absence of a critical diffusion coefficient for the one-dimensional case, contrasting with known results in higher dimensions.

Findings

No critical diffusion coefficient exists in 1D.

Solutions with small mass exist globally.

Large mass solutions can blow up in finite time.

Abstract

We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.