Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

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

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.

Cross Diffusion Preventing Blow-Up in the Two-Dimensional Keller-Segel Model

Abstract. A (Patlak-) Keller-Segel model in two space dimensions with an additional crossdiffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross diffusion, the global existence of ...

The Stability and Dynamics of a Spike in the One-Dimensional Keller-Segel model

In the limit of a large mass M À 1, and on a finite interval of length 2L, an equilibrium spike solution to the classical Keller-Segel chemotaxis model with a linear chemotactic function is constructed asymptotically. By calculating an asymptotic formula for the translational eigenvalue for M À 1, it is shown that the equilibrium spike solution is unstable to translations of the spike profile. ...