Related papers: A uniform controllability result for the Keller-Se…
This paper deals with convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the (Navier-)Stokes equations in the framework of the full model \begin{eqnarray*} \left\{ \begin{array}{lcl} \, \, \partial_t…
In this paper we consider quasilinear Keller-Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite…
We consider a parabolic-parabolic Keller-Segel system of chemotaxis model with singular sensitivity $u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)$, $v_t=k\Delta v-v+u$ under homogeneous Neumann boundary conditions in a smooth bounded…
We consider a system coupling the parabolic-parabolic Keller-Segel equations to the in- compressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up…
A parabolic-parabolic (Patlak-) Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy…
We study the solutions of the two-dimensional Keller-Segel system describing chemotaxis. The Keller-Segel system as well as the properties of the blow-up set has been extensively studied. In this paper we obtain generalized solutions for…
In this paper, we shall study the parabolic-elliptic Keller-Segel system on the Poincar{\'e} disk model of the 2D-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this…
We construct radial self-similar solutions of the, so called, minimal parabolic-elliptic Keller--Segel model in several space dimensions with radial, nonnegative initial conditions with are below the Chandrasekhar solution -- the singular…
This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=\Delta (\gamma (v) u ) &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] v_t=\Delta v - v + u &\mathrm{in}\ \Omega\times(0,\infty),…
We consider a parabolic-elliptic Keller-Segel type system, which is related to a simplified model of chemotaxis. Concerning the maximal range of existence of solutions, there are essentially two kinds of results: either global existence in…
This paper deals with the null controllability of a coupled parabolic system, which is Kuramoto-Sivashinsky-Korteweg-de Vries equation coupled with heat equation through first order derivative. More precisely, we prove the null…
We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a non-differentiable term with the measure-norm of…
In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean-field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the…
This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel in the full space. We derive a critical mass threshold below which global existence is ensured. Using carefully energy…
In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions…
Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the…
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…
The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or "free energy") solutions associated to an initial datum with finite mass $M\textless{} 8\pi$, finite second…
We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity…
The aim of this paper is to perform a Stackelberg strategy to control parabolic equations. We have one control, \textit{the leader}, that is responsible for a null controllability property; additionally, we have a control \textit{the…