Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

TL;DR

This paper investigates the existence of finite mass self-similar blow-up solutions in a chemotaxis system with nonlinear diffusion, revealing a critical mass threshold and solutions in higher dimensions.

Contribution

It demonstrates the existence of finite mass self-similar blow-up solutions in dimensions greater than three for a chemotaxis system with nonlinear diffusion, extending previous results.

Findings

Existence of finite mass self-similar blow-up solutions in dimensions >3

Identification of a critical mass threshold for global existence and blow-up

Differentiation of behavior from the 2D case

Abstract

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all the solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d?3$.