Finite-time blow-up in the Cauchy problem of a Keller-Segel system with logistic source

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 ...

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...

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

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.

Exact Criterion for Global Existence and Blow Up to a Degenerate Keller-Segel System

A degenerate Keller-Segel system with diffusion exponent m with 2n n+2 < m < 2 − 2 n in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on L 2n n+2 norm of the solution play important roles in our analysis. These estimates are closely related to the optimal constant in the HardyLittlewoodSobolev inequality. In the case of in...