Related papers: Blow-up, concentration phenomenon and global exist…
We consider a Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen…
A class of parabolic-parabolic Keller-Segel systems with degenerate diffusion and volume filling is studied in a bounded domain subject to no-flux boundary conditions. The equations are derived from a multiphase fluid model. The interplay…
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…
We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data…
The focus of this paper is on solutions to a two-dimensional Keller-Segel system containing sub-logistic sources. We show that the presence of sub-logistic terms is adequate to prevent blow-up phenomena even in strongly degenerate…
We consider nonnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel system \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\Delta u-\nabla \cdot \big(u\nabla v\big),\\ &0=\Delta v -\mu + u , \\…
We consider the parabolic-elliptic Keller-Segel system in spatial dimensions $d\geq3$, which corresponds to the mass supercritical case. Some solutions become singular in finite time, an important example being backward self-similar…
The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [46], mass conservation, positivity and energy decay were proved for the proposed scheme,…
The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the…
We investigate blow-up properties for the initial-boundary value problem of a Keller-Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller-Segel model, we first…
We study the global strong solutions to a 3-dimensional parabolic-hyperbolic Keller-Segel model with initial data close to a stable equilibrium with perturbations belonging to $L^2(\mathbb R^3)\times H^1(\mathbb{R}^3)$. We obtain global…
The paper should be viewed as complement of an earlier result in [8]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there…
A degenerate Keller-Segel system with diffusion exponent $2n/(n+2)<m<2-\frac{2}{n}$ in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on $L^{2n/(n+2)}$ norm of the…
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 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…
In this paper, we obtain the exact blow-up profiles of solutions of the Keller-Segel-Patlak system in the space with dimensions $N\ge 3$, which solves an open problem proposed by P. Souplet and M. Winkler in 2019. To establish this…
In Talay and Tomasevic [20] we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cutoff. It involved an original type of McKean-Vlasov interaction kernel which involved all the past time…
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…
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 study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot…