Global existence of solutions to a parabolic-elliptic chemotaxis system with critical degenerate di ffusion

TL;DR

This paper proves the global existence of weak solutions for a chemotaxis model with critical degenerate diffusion, under small initial mass, extending previous blow-up results and establishing uniqueness under regularity conditions.

Contribution

It establishes the global existence of solutions for a chemotaxis system with critical degenerate diffusion, filling a gap in the understanding of solution behavior for large initial mass.

Findings

Global weak solutions exist under small initial mass.

Uniqueness of solutions is proved with higher regularity.

Addresses finite-time blow-up for large masses.

Abstract

This paper is devoted to the analysis of non-negative solutions for a degenerate parabolic-elliptic Patlak-Keller-Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, there by completing previous results on nite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.