We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This mode...

An equation to describe nearly 1D traveling-waves patterns is put forward in the form of a dispersive generalization of the Newell-Whitehead-Segel equation. Transverse stability of plane waves is shown to be drastically altered by the dispersion. A necessary transverse Benjamin-Feir stability condition is obtained. If it is met, a quarter of the plane-wave existence band is unstable, while thre...

We give a simple proof, relying on two-particles moment computation, that there exists global weak solution to the $ 2 $-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure f_0 such f_0( {\mathbb{R}}^2)< 8 \pi $.

In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of pertubations (i.e. in divergence form), the equation has global weak solution small initial data. Furthermore, if noise is not form, show finite time blowup (with nonzero probability) any The results on continuous dependence solutions perturbations, alongside ex...

Abstract. Variational steepest descent approximation schemes for the modified Patlak-KellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recov...

The Keller-Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing up solutions for large enough initial conditions in dimensions d ≥ 2, but all the solutions are regular in one dimension; a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions...