# 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} }

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…

## 25 Citations

### Parabolic elliptic type Keller-Segel system on the whole space case

- 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…

### Supercritical degenerate parabolic–parabolic Keller–Segel system: existence criterion given by the best constant in Sobolev’s inequality

- MathematicsZeitschrift 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…

### Sharp conditions on global existence and blow-up in a degenerate two-species and cross-attraction system

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- 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…

### Global existence versus blow-up in a high dimensional Keller–Segel equation with degenerate diffusion and nonlocal aggregation

- Mathematics
- 2015

### A Note on L∞$L^{\infty}$-Bound and Uniqueness to a Degenerate Keller-Segel Model

- 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$\frac {2n}{n+2}< m<2-\frac{2}{n}$…

### The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime

- 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)…

### A degenerate $p$-Laplacian Keller-Segel model

- 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 <…

### A generalized Sz. Nagy inequality in higher dimensions and the critical thin film equation

- 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…

### Compactly supported stationary states of the degenerate Keller-Segel system in the diffusion-dominated regime

- 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…

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