The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π , is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691–721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is full...

We introduce stochastic models of chemotaxis generalizing the deterministic KellerSegel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean’s approach, we derive the exact kinetic equation satisfied by the density distribution of cells. In the mean field limit where statistical correlations between cell...

We consider here a recently developed upwind-difference potentials method for chemotaxis models [8]. The introduced scheme can be used to approximate problems in complex geometries. The chemotaxis model under consideration is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reactiondiffusion equation for the chemoattractant concent...

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller-Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of t...

We consider a rather general class of convection–diffusion equations, involving dissipation (of possibly fractional order) which competes with quadratic nonlinearities on the regularity of the overall equation. This includes as prototype models, Burgers’ equation, the Navier–Stokes equations, the surface quasigeostrophic equations and the Keller–Segel model for chemotaxis. Here we establish a P...