Global convergence of a kinetic model of chemotaxis to a perturbed Keller–Segel model

We consider a class of kinetic models of chemotaxis with two positive non-dimensional parameters coupled to a parabolic equation of the chemo-attractant. If both parameters are set equal zero, we have the classical Keller–Segel model for chemotaxis. We prove global existence of solutions of this two-parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero. In one case, we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed Keller–Segel model with global existence of solutions.     

Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data

The goal of this paper is to exhibit a critical mass phenomenon occurring in a model for cell self-organization via chemotaxis. The very well-known dichotomy arising in the behavior of the macroscopic Keller–Segel system is derived at the kinetic level, being closer to microscopic features. Indeed, under the assumption of spherical symmetry, we prove that solutions with initial data of large mass blow-up in finite time, whereas solutions with initial data of small mass do not. Blow-up is the consequence of a momentum computation and the existence part is derived from a comparison argument. Spherical symmetry is crucial within the two approaches. We also briefly investigate the drift-diffusion limit of such a kinetic model. We recover partially at the limit the Keller–Segel criterion for blow-up, thus arguing in favour of a global link between the two models.     

Shock formation in a chemotaxis model

In this paper, we establish the existence of shock solutions for a simplified version of the Othmer–Stevens chemotaxis model (SIAM J. Appl. Math. 1997; 57 :1044–1081). The existence of these shock solutions was suggested by Levine and Sleeman (SIAM J. Appl. Math. 1997; 57 :683–730). Here, we consider the general Riemann problem and derive the shock curves in parameterized forms. By studying the travelling wave solutions, we examine the shock structure for the chemotaxis model and prove that the travelling wave speed is identical to the shock speed. Moreover, we explicitly derive an entropy–entropy flux pair to prove the uniqueness of the weak shock solutions. Some discussion is given for further study. Copyright © 2007 John Wiley & Sons, Ltd.     

A chemotaxis model motivated by angiogenesis

We consider a simple model arising in modeling angiogenesis and more specifically the development of capillary blood vessels due to an exogenous chemo-attractive signal (solid tumors for instance). It is given as coupled system of parabolic equations through a nonlinear transport term. We show that, by opposition to some classical chemotaxis model, this system admits a positive energy. This allows us to develop an existence theory for weak solutions. We also show that, in two dimensions, this system admits a family of self-similar waves. To cite this article: L. Corrias et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Preventing blow-up in a chemotaxis model

This paper deals with a chemotactic model of Keller-Segel type. The main feature of the Keller-Segel model is the possibility of blow-up of solutions in a finite time. To eliminate the possibility of blow-up a modified version of Keller-Segel model is introduced. The blow-up control relies on the presence of a pressure function, which increases faster than a logarithm for high enough cells densities: for such a pressure function the solutions cannot blow up in a finite time. Some other conditions are introduced to ensure the global boundedness.     

Dynamics of a bio-reactor model with chemotaxis

Long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio-reactors with chemotaxis will be studied. In particular, we show that the dynamical system possesses a global attractor and that it is strongly uniformly persistent if the trivial steady state is unstable. Using a result of Smith and Waltman on perturbation of global attractors, we also show that the positive steady state is unique and globally attracting.     

Diffusion-dominated asymptotics of solution to chemotaxis model

The paper contains results on the asymptotic behavior, as t → +∞, of small solutions to simplified Keller–Segel problem modeling chemotaxis in the whole space \mathbb R²{\mathbb R^2}. We prove that the multiple of the heat kernel is a surprisingly good approximation of solutions.     

On the time-fractional Keller-Segel model for chemotaxis

We consider the time-fractional Keller-Segel system of order α∈(0,1). Interesting properties of solutions are highlighted, like regularity and large time behavior in Lebesgue spaces, which depend on the fractional exponent α.     

Finite volume methods for degenerate chemotaxis model

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection-diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.     

Global attractor for a density-dependent sensitivity chemotaxis model

In this paper, a chemotaxi model with reproduction term in a bounded domain Ω ⊂ Rⁿ is discussed. The existence of a global-in-time solution and a global attractor for this model are obtained.     

Blow-up and decay criteria for a model of chemotaxis

For a system of equations introduced by Jäger and Luckhaus (1992) [6] as a model of chemotaxis, the questions of blow-up and global existence criteria are investigated. Specifically, for a convex region, a lower bound for the blow-up time is derived if the solution blows up, and explicit criteria to ensure non-blow-up are determined.     

On convergence to equilibria for the Keller-Segel chemotaxis model

We show that any global-in-time bounded solution to the Keller-Segel chemotaxis model converges to a single equilibrium as time tends to infinity. The proof is based on a generalized version of the Lojasiewicz-Simon theorem.     

Lower bounds for blow-up in a model of chemotaxis

Many special cases of the classical Keller–Segel system for modeling chemotaxis have been investigated in the literature, and typically the solution of the governing equations will blow up at some finite time. However, the question of establishing lower bounds for this blow-up time has been largely ignored. This paper derives such a lower bound in a parabolic–parabolic model in both ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$.     

A stochastic Keller–Segel model of chemotaxis

We introduce stochastic models of chemotaxis generalizing the deterministic Keller–Segel 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 cells are neglected, we recover the Keller–Segel model governing the smooth density field. We also consider hydrodynamic and kinetic models of chemotaxis that take into account the inertia of the particles and lead to a delay in the adjustment of the velocity of cells with the chemotactic gradient. We make the connection with the Cattaneo model of chemotaxis and the telegraph equation.