Related papers: Critical mass for a Patlak-Keller-Segel model with…
This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass…
For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is…
We study radial solutions in a ball of $\mathbb{R}^N$ of a semilinear, parabolic-elliptic Patlak-Keller-Segel system with a nonlinear sensitivity involving a critical power. For $N = 2$, the latter reduces to the classical linear model,…
The L^1-critical parabolic-elliptic Patlak-Keller-Segel system is a classical model of chemotactic aggregation in micro-organisms well-known to have critical mass phenomena. In this paper we study this critical mass phenomenon in the…
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 consider a class of $L^1$ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at…
This paper is devoted to the analysis of non-negative solutions for a degenerate parabolic-elliptic Patlak-Keller-Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to…
In this paper, we study the following Patlak-Keller-Segel model with $p$-Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &\rho _t=\nabla \cdot \left( \left| \nabla \rho \right|^{p-2}\nabla \rho \right) -\chi \nabla \cdot \left(…
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…
In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no…
It is known that, for the parabolic-elliptic Keller-Segel system with critical porous-medium diffusion in dimension $\RR^d$, $d \ge 3$ (also referred to as the quasilinear Smoluchowski-Poisson equation), there is a critical value of the…
We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of…
The qualitative study of solutions to the coupled parabolic-elliptic chemotaxis system with nonlinear diffusion for two species will be considered in the whole Euclidean space $\mathbb{R}^d$ ($d\geq 3$). It was proven in \cite{CK2021-ANA}…
In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of…
In this paper we consider the parabolic-elliptic Patlak-Keller-Segel models in $\mathbb T^d$ with $d=2,3$ with the additional effect of advection by a large shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions…
We carry on our studies related to the fully parabolic quasilinear Keller-Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of radially symmetric solutions to the quasilinear…
In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in $\mathbb{T}\times\mathbb{R}$ with advection by a large strictly monotone shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with…
We show that the one-dimensional fully parabolic Keller-Segel system with nonlinear diffusion possesses global-in-time solutions, provided the nonlinear diffusion is equal to (1+u)^{-\alpha}, for \alpha < 1, independently on the volume of…
This paper is concerned with the global boundedness and blowup of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility…
We study the global in time existence and long time asymptotics of solutions to the parabolic-elliptic Patlak-Keller-Segel system for the multi-species populations in the whole Euclidean space $\mathbb{R}^2.$ We prove that at the borderline…