Finite time blow up and concentration phenomena for a solution to drift-diffusion equations in higher dimensions

We show the finite time blow up of a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If the initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not exist globally in time and blows up in a finite time for the scaling critical space. Besides there exists a concentration point such that the solution exhibits the concentration in the critical norm. This type of blow up was observed in the scaling critical two dimensions. The proof is based on the profile decomposition and the Shannon inequality in the weighted space

Critical Keller-Segel meets Burgers on S1{\mathbb S}^1: large-time smooth solutions

We show that solutions to the parabolic-elliptic Keller-Segel system on ${\mathbb S}^1$ with critical fractional diffusion $(-\Delta)^\frac{1}{2}$ remain smooth for any initial data and any positive time. This disproves, at least in the periodic setting, the large-data-blowup conjecture by Bournaveas and Calvez. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the method of moduli of continuity by Kiselev, Nazarov and Shterenberg. over a setting where the considered equation has no scaling. This auxiliary result may be interesting by itself. Finally, we study the asymptotic behavior of global solutions, improving the existing results.Comment: 17 page

Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system

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Numerical simulation of the dynamics of molecular markers involved in cell polarisation

A cell is polarised when it has developed a main axis of organisation through the reorganisation of its cytosqueleton and its intracellular organelles. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signaling molecules ... In this work, we study mathematical models for cell polarisation. These models are based on nonlinear convection-diffusion equations. The nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome.Comment: 15 page