# Exact criterion for global existence and blow up to a degenerate Keller-Segel system

@article{Chen2013ExactCF,
title={Exact criterion for global existence and blow up to a degenerate Keller-Segel system},
author={Li Chen and Jinhuan Wang},
journal={Documenta Mathematica},
year={2013}
}
• Published 1 November 2013
• Mathematics
• Documenta Mathematica
A degenerate Keller-Segel system with diffusion exponent m with 2n n+2 < m < 2 − 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 solution play important roles in our analysis. These estimates are closely related to the optimal constant in the Hardy- Littlewood- Sobolev inequality. In the case of initial free energy less than a universal constant which depends on the inverse of total mass, there…
• Mathematics
• 2015
This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is
• Mathematics
Zeitschrift für angewandte Mathematik und Physik
• 2019
This article presents a relationship between the sharp constant of the Sobolev inequality and the initial criterion to the global existence of degenerate parabolic–parabolic Keller–Segel system with
• Mathematics
• 2021
Abstract We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ 3 and find two critical curves intersecting at one point which separate the global existence and
• Mathematics
• 2016
In this note we establish the uniform L∞$L^{\infty}$-bound for the weak solutions to a degenerate Keller-Segel equation with the diffusion exponent 2nn+2<m<2−2n$\frac {2n}{n+2}< m<2-\frac{2}{n}$
• Mathematics
• 2017
We consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular)
• Mathematics
• 2016
This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < • Mathematics • 2016 In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions d⩾1 J(h):=(∫Rd|h| dx)a−1∫Rd|∇h|2 • Mathematics • 2016 We first show the existence of unique global minimizer of the free energy for all masses associated to a nonlinear diffusion version of the classical Keller-Segel model when the diffusion dominates ## References SHOWING 1-10 OF 22 REFERENCES • Mathematics • 2008 This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear • Mathematics • 2006 The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion • Mathematics SIAM J. Math. Anal. • 2012 It is proved that there is a global weak solution to the degenerate diffusion Patlak–Keller–Segel system and if the initial data are strictly below$U_{\lambda,0}(x)$for some$\lambda$, then the solution vanishes in$L^1_{loc}$as$t\to\infty\$ as time goes to infinity.
• Mathematics
• 2007
We analyze the two‐dimensional parabolic‐elliptic Patlak‐Keller‐Segel model in the whole Euclidean space ℝ2. Under the hypotheses of integrable initial data with finite second moment and entropy, we
The partial differential equation of the random walk problem with persistence of direction and external bias is derived. By persistence of direction or internal bias we mean that the probability a