We investigate nonnegativity of exact and numerical solutions to a generalized Keller– Segel model. This model includes the so-called ‘‘minimal’’ Keller–Segel model, but can cover more general chemistry. We use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative. We then derive numerical methods, using finite element techni...

In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. I...

We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The describes dynamics oxygen and bacteria densities evolving within a fluid. establish local well-posedness in Sobolev spaces for partially inviscid fully cases. In latter, additional assumptions on initial data are required when either or density touches zero. Even though satisfies maximum p...

In this paper we study a version of the Keller–Segel model where the chemotactic cross-diffusion depends on both the external signal and the local population density. A parabolic quasi-linear strongly coupled system follows. By incorporation of a population-sensing (or “quorum-sensing”) mechanism, we assume that the chemotactic response is switched off at high cell densities. The response to hi...

In this paper we study the controllability of the Keller-Segel system approximating its parabolic-elliptic version. We show that this parabolic system is locally uniform controllable around a constant solution of the parabolic-elliptic system when the control is acting on the component of the chemical. Résumé. Dans cet article, nous étudions la contrôlabilité du système de Keller-Segel qui appr...

We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions ̺λ, λ > 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R ...

Abstract We consider a singular limit problem of the Cauchy for Patlak–Keller–Segel equation in scaling critical function space. It is shown that solution to system space involving class bounded mean oscillations converges drift-diffusion parabolic-elliptic type (simplified Keller–Segel model) strongly as relaxation time parameter $$\tau \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3....