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On the self-similar stability of the parabolic-parabolic Keller-Segel equation

Frank Alvarez Borges, Kleber Carrapatoso, Stéphane Mischler

TL;DR

The paper studies the parabolic-parabolic Keller-Segel system in $\mathbb{R}^2$ using self-similar variables and a small time-scale parameter $\varepsilon>0$, focusing on the exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. A perturbation argument at the first component of the linearized operator together with a purely semigroup analysis provides linear and nonlinear exponential decay in weighted Sobolev spaces without assuming radial symmetry of the initial data. The authors develop detailed dissipativity and spectral analyses for the diagonal blocks $\mathcal{L}_{1,1}$ and $\mathcal{L}_{2,2}$, derive robust semigroup decay for the full linearized system, and implement a fixed-point scheme to obtain global well-posedness and exponential stability for small initial data. Consequences include the absence of concentration and diffusion-dominated long-time behavior with a self-similar profile, extending prior radially symmetric results to general initial data and connecting the parabolic-parabolic dynamics to the parabolic-elliptic limit.

Abstract

We consider the parabolic-parabolic Keller-Segel equation in the plane and prove the nonlinear exponential stability of the self-similar profile in a quasi parabolic-elliptic regime. We first perform a perturbation argument in order to obtain exponential stability for the semigroup associated to part of the first component of the linearized operator, by exploiting the exponential stability of the linearized operator for the parabolic-elliptic Keller-Segel equation. We finally employ a purely semigroup analysis to prove linear, and then nonlinear, exponential stability of the system in appropriated functional spaces.

On the self-similar stability of the parabolic-parabolic Keller-Segel equation

TL;DR

The paper studies the parabolic-parabolic Keller-Segel system in $\mathbb{R}^2$ using self-similar variables and a small time-scale parameter $\varepsilon>0$, focusing on the exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. A perturbation argument at the first component of the linearized operator together with a purely semigroup analysis provides linear and nonlinear exponential decay in weighted Sobolev spaces without assuming radial symmetry of the initial data. The authors develop detailed dissipativity and spectral analyses for the diagonal blocks $\mathcal{L}_{1,1}$ and $\mathcal{L}_{2,2}$, derive robust semigroup decay for the full linearized system, and implement a fixed-point scheme to obtain global well-posedness and exponential stability for small initial data. Consequences include the absence of concentration and diffusion-dominated long-time behavior with a self-similar profile, extending prior radially symmetric results to general initial data and connecting the parabolic-parabolic dynamics to the parabolic-elliptic limit.

Abstract

We consider the parabolic-parabolic Keller-Segel equation in the plane and prove the nonlinear exponential stability of the self-similar profile in a quasi parabolic-elliptic regime. We first perform a perturbation argument in order to obtain exponential stability for the semigroup associated to part of the first component of the linearized operator, by exploiting the exponential stability of the linearized operator for the parabolic-elliptic Keller-Segel equation. We finally employ a purely semigroup analysis to prove linear, and then nonlinear, exponential stability of the system in appropriated functional spaces.
Paper Structure (11 sections, 18 theorems, 205 equations)

This paper contains 11 sections, 18 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Estimates over $Q$ and $P$
  3. Functional inequalities
  4. Estimates for $\mathcal{L}_{1,1}$
  5. Dissipativity estimates related to $\mathcal{L}_{1,1}$.
  6. Splitting of the operator $\mathcal{L}_{1,1}$
  7. Spectral analysis of $\mathcal{L}_{1,1}$
  8. Semigroup decay estimates for $\mathcal{L}_{1,1}$
  9. Estimates for $\mathcal{L}_{2,2}$
  10. Semigroup estimates for the linearized system
  11. Proof of the nonlinear stability theorem

Key Result

Theorem 1.1

Let us fix $\mu \in (0,\infty)$, $k > 3$ and $s \in (0,1)$. There are $\varepsilon_0 , \eta_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ and any initial data $(g_0, w_0 )\in \mathcal{X}$ with $\langle\!\langle g_0 \rangle\!\rangle = 0$ and $\| (g_0, w_0) \|_{\mathcal{X}} \le \eta_0$ Moreover, for any $\lambda \in (0,\mu(1-s))$, we have the exponential decay estimate

Theorems & Definitions (33)

  • Theorem 1.1
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['UniConChap7']}
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:UnifBddQP']}
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lem:eps0']}
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:borneKgHs']}
  • Lemma 3.2
  • ...and 23 more