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

Volume effects in the Keller-Segel model: energy estimates preventing blow-up

We obtain a priori estimates for the classical chemotaxis model of Patlak, Keller and Segel when a nonlinear diffusion or a nonlinear chemosensitivity is considered accounting for the finite size of the cells. We will show how entropy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solutions.

The one-dimensional Keller-Segel model with fractional diffusion of cells

We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent 0 < α ≤ 2. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when α < 1 and the initial confi...

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.

New Interior Penalty Discontinuous Galerkin Methods for the Keller-Segel Chemotaxis Model

We develop a family of new interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbol...