Asymptotic Behavior for Critical Patlak-Keller-Segel model and an Repulsive-Attractive Aggregation Equation

@article{Yao2011AsymptoticBF,
  title={Asymptotic Behavior for Critical Patlak-Keller-Segel model and an Repulsive-Attractive Aggregation Equation},
  author={Yao Yao},
  journal={arXiv: Analysis of PDEs},
  year={2011}
}
  • Yao Yao
  • Published 2011
  • Mathematics
  • arXiv: Analysis of PDEs
In this paper we study the long time asymptotic behavior for a class of diffusion-aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum-principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak-Keller-Segel problem with critical power $m=2-2/d$, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial… Expand
The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime
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)Expand
Equilibria of homogeneous functionals in the fair-competition regime
We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular/non-singular kernel leading to variants of theExpand
Uniqueness of stationary states for singular Keller–Segel type models
We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than NewtonianExpand
Parabolic elliptic type Keller-Segel system on the whole space case
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 isExpand
Aggregation Equation with Degenerate Diffusion
Recently, there has been a growing interest in the use of nonlocal partial differential equation (PDE) to model biological and physical phenomena. In this dissertation, we study the behavior ofExpand
Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits
Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equationsExpand
Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition
We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initialExpand
Blow-Up Phenomena for Gradient Flows of Discrete Homogeneous Functionals
We investigate gradient flows of some homogeneous functionals in $$\mathbb R^N$$RN, arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on theExpand
H\"older regularity and Uniqueness theorem on weak solutions to the degenerate Keller-Segel system
In this paper, we present local H\"older estimates for the degenerate Keller-Segel system \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} below in the range of $m>1$ and $q>1$ before aExpand
A GRADIENT FLOW APPROACH TO THE KELLER-SEGEL SYSTEMS (Progress in Variational Problems : Variational Problems Interacting with Probability Theories)
These notes are dedicated to recent global existence and regularity results on the parabolic-elliptic Keller-Segel model in dimension 2, and its generalisation with nonlinear diffusion in higherExpand
...
1
2
...

References

SHOWING 1-10 OF 37 REFERENCES
Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions
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-linearExpand
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
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-diffusionExpand
Asymptotic Estimates for the Parabolic-Elliptic Keller-Segel Model in the Plane
We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8 π after a properExpand
Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2
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, weExpand
The Patlak-Keller-Segel Model and Its Variations: Properties of Solutions via Maximum Principle
TLDR
This paper investigates qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations with challenge of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. Expand
Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model
Abstract The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservativeExpand
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion
Recently, there has been a wide interest in the study of aggregation equations and Patlak–Keller–Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is theExpand
Global Existence and Finite Time Blow-Up for Critical Patlak-Keller-Segel Models with Inhomogeneous Diffusion
TLDR
The primary issue is how, if possible, one localizes the presence of the inhomogeneity in the nonlocal term of the Patlak--Keller--Segel models with spatially varying diffusivity of the chemoattractant. Expand
Optimal critical mass in the two dimensional Keller–Segel model in R2
Abstract The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusionExpand
Asymptotic Dynamics of Attractive-Repulsive Swarms
TLDR
An analytical upper bound is derived for the finite blow-up time after which the solution forms one or more $\delta$-functions of the conservation equation. Expand
...
1
2
3
4
...