• Corpus ID: 208267949

Spectral analysis for singularity formation of the two dimensional Keller-Segel system.

@article{Collot2019SpectralAF,
  title={Spectral analysis for singularity formation of the two dimensional Keller-Segel system.},
  author={Charles Collot and Tej-eddine Ghoul and Nader Masmoudi and Van Tien Nguyen},
  journal={arXiv: Analysis of PDEs},
  year={2019}
}
We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as… 
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References

SHOWING 1-10 OF 31 REFERENCES
Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System
We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state
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-diffusion
Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions
  • J. Velázquez
  • Mathematics, Computer Science
    SIAM J. Appl. Math.
  • 2004
TLDR
This paper analyzes the precise way in which the regularization introduced in the Keller--Segel system stops the aggregation process and yields the formation of concentration regions.
Logarithmic scaling of the collapse in the critical Keller-Segel equation
A reduced Keller-Segel equation (RKSE) is a parabolic-elliptic system of partial differential equations which describes bacterial aggregation and the collapse of a self-gravitating gas of brownian
On the Stability of Type I Blow Up For the Energy Super Critical Heat Equation
We consider the energy super critical semilinear heat equation $$\partial_t u=\Delta u+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.$$ We first revisit the construction of radially symmetric backward self
Symmetrization Techniques on Unbounded Domains: Application to a Chemotaxis System on RN
The authors study the parabolic-elliptic system on RN: ∂u/∂t=∇⋅(∇u−χu∇v), 0=Δv−γv+αu, u(0,⋅)=u0, a version of the mathematical model of chemotaxis proposed by Keller and Segel. A differential
Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model
We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions $\rhohls$, $\lambda>0$, with thick tails
On spectra of linearized operators for Keller–Segel models of chemotaxis
Abstract We consider the phenomenon of collapse in the critical Keller–Segel equation (KS) which models chemotactic aggregation of micro-organisms underlying many social activities, e.g. fruiting
Singularity patterns in a chemotaxis model
The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up
On strongly anisotropic type II blow up
We consider the energy super critical 4 dimensional semilinear heat equation $$\partial_tu=\Delta u+|u|^{p-1}u, \ \ x\in \Bbb R^4, \ \ p>5.$$ Let $\Phi(r)$ be a three dimensional radial self similar
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