ERRATUM: Suppression of Blow-Up in Patlak--Keller--Segel Via Shear Flows

Global Existence and Finite Time Blow-Up for Critical Patlak-Keller-Segel Models with Inhomogeneous Diffusion

The L-critical parabolic-elliptic Patlak-Keller-Segel system is a classical model of chemotactic aggregation in micro-organisms well-known to have critical mass phenomena [10, 8]. In this paper we study this critical mass phenomenon in the context of Patlak-Keller-Segel models with spatially varying diffusivity and decay rate of the chemo-attractant. The primary tool for the proof of global exi...

Boundedness vs. blow-up in the Keller-Segel system

The fully parabolic Keller-Segel chemotaxis system { ut = ∆u−∇ · (u∇v), x ∈ Ω, t > 0, vt = ∆v − v + u, x ∈ Ω, t > 0, is considered under homogeneous Neumann boundary conditions in bounded domains Ω ⊂ R, n ≥ 1. We demonstrate rigorous analytical techniques which can be used to identify situations when solutions either remain bounded, or exhibit a blow-up phenomenon. In the latter case, which is ...

Blow up of solutions to generalized Keller–Segel model

The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.

The two-dimensional Keller-Segel model after blow-up

The Patlak-Keller-Segel Model and Its Variations: Properties of Solutions via Maximum Principle

In this paper we investigate qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concen...