Differential and Integral Equations

Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type

Takayoshi Ogawa

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Abstract

We discuss the global behavior of the weak solution of the Keller-Segel system of degenerate type. Asymptotic stability of the Barenblatt-Pattle solution and its convergence rate for the decaying weak solution in $L^1({\mathbb R}^n)$ is shown for the degenerated case $1 <{\alpha} < 2-\frac{2}{n}$. The method is based on the techniques applied to the Fokker-Plank equation due to Carrillo-Toscani [8] deriving from the explicit time decay of the free energy functional and some new estimates for the nonlinear interaction involving the critical type Sobolev inequality. We give the rigorous justification of those procedures via some approximating procedures.

Article information

Source
Differential Integral Equations, Volume 21, Number 11-12 (2008), 1113-1154.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1355502296

Mathematical Reviews number (MathSciNet)
MR2482499

Zentralblatt MATH identifier
1224.35229

Subjects
Primary: 35M31: Initial value problems for systems of mixed type
Secondary: 35B35: Stability 35K65: Degenerate parabolic equations

Citation

Ogawa, Takayoshi. Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type. Differential Integral Equations 21 (2008), no. 11-12, 1113--1154. https://projecteuclid.org/euclid.die/1355502296


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