The Keller-Segel system of parabolic-parabolic type with initial data in weak Ln/2(ℝn) and its application to self-similar solutions

Hideo Kozono, Yoshie Sugiyama

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

We shall show the existence of a global strong solution to the semilinear Keller-Segel system in ℝn, n ≥ 3 of parabolic-parabolic type with small initial data u0 ∈ Lw n/2 (ℝn) and v0 ∈ BMO. Our method is based on the perturbation of linearization together with the Lp -L q-estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.

Original languageEnglish
Pages (from-to)1467-1500
Number of pages34
JournalIndiana University Mathematics Journal
Volume57
Issue number4
DOIs
Publication statusPublished - 2008
Externally publishedYes

Fingerprint

Self-similar Solutions
Strong Solution
Heat Semigroup
Smoothing Effect
Fractional Powers
Laplace Operator
Semilinear
Linearization
Perturbation
Estimate

Keywords

  • BMO space
  • Heat semigroup
  • Keller-Segel system
  • Self-similar solution
  • Weak L -space

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

@article{bed8b9e0278d4e498084ba873f266245,
title = "The Keller-Segel system of parabolic-parabolic type with initial data in weak Ln/2(ℝn) and its application to self-similar solutions",
abstract = "We shall show the existence of a global strong solution to the semilinear Keller-Segel system in ℝn, n ≥ 3 of parabolic-parabolic type with small initial data u0 ∈ Lw n/2 (ℝn) and v0 ∈ BMO. Our method is based on the perturbation of linearization together with the Lp -L q-estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.",
keywords = "BMO space, Heat semigroup, Keller-Segel system, Self-similar solution, Weak L -space",
author = "Hideo Kozono and Yoshie Sugiyama",
year = "2008",
doi = "10.1512/iumj.2008.57.3316",
language = "English",
volume = "57",
pages = "1467--1500",
journal = "Indiana University Mathematics Journal",
issn = "0022-2518",
publisher = "Indiana University",
number = "4",

}

TY - JOUR

T1 - The Keller-Segel system of parabolic-parabolic type with initial data in weak Ln/2(ℝn) and its application to self-similar solutions

AU - Kozono, Hideo

AU - Sugiyama, Yoshie

PY - 2008

Y1 - 2008

N2 - We shall show the existence of a global strong solution to the semilinear Keller-Segel system in ℝn, n ≥ 3 of parabolic-parabolic type with small initial data u0 ∈ Lw n/2 (ℝn) and v0 ∈ BMO. Our method is based on the perturbation of linearization together with the Lp -L q-estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.

AB - We shall show the existence of a global strong solution to the semilinear Keller-Segel system in ℝn, n ≥ 3 of parabolic-parabolic type with small initial data u0 ∈ Lw n/2 (ℝn) and v0 ∈ BMO. Our method is based on the perturbation of linearization together with the Lp -L q-estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.

KW - BMO space

KW - Heat semigroup

KW - Keller-Segel system

KW - Self-similar solution

KW - Weak L -space

UR - http://www.scopus.com/inward/record.url?scp=54049107301&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=54049107301&partnerID=8YFLogxK

U2 - 10.1512/iumj.2008.57.3316

DO - 10.1512/iumj.2008.57.3316

M3 - Article

AN - SCOPUS:54049107301

VL - 57

SP - 1467

EP - 1500

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 4

ER -