# 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…

## 4 Citations

Infinite time blow-up in the Keller-Segel system: existence and stability

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The simplest version of the parabolic-elliptic Patlak-Keller-Segel system in the two-dimensional Euclidean space has an 8π critical mass which corresponds to the exact threshold between finite-time…

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

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Abstract. We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the…

Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

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In this paper, we consider the standard semilinear heat equation ∂tu = ∆u+ |u|p−1u, p > 1. The determination of the (believed to be) generic blowup profile is well-established in the literature, with…

## References

SHOWING 1-10 OF 31 REFERENCES

Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System

- Mathematics, Physics
- 2019

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

- 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…

Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions

- Mathematics, Computer ScienceSIAM J. Appl. Math.
- 2004

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

- Mathematics, Physics
- 2013

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

- MathematicsMemoirs of the American Mathematical Society
- 2019

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

- Mathematics
- 1998

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

- Physics, Mathematics
- 2010

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

- Mathematics, Physics
- 2012

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

- Mathematics
- 1996

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

- Physics, Mathematics
- 2017

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…