Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

Hideo Kozono, Yoshie Sugiyama

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u0 ∈ L1 (ℝ2). The local existence time is characterized for u 0 ∈ L1 ∩ Lq* (ℝ2) with 1 < q* < 2. Next, we prove the finite time blow-up of strong solution under the assumption ∥u0L1 > 8π and ∥x|2u0L1 < 1/γ·g (∥u0L1/8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

Original languageEnglish
Pages (from-to)353-378
Number of pages26
JournalJournal of Evolution Equations
Volume8
Issue number2
DOIs
Publication statusPublished - 2008 May
Externally publishedYes

Fingerprint

Finite Time Blow-up
Blow-up of Solutions
Local Existence
Mild Solution
Blow-up Rate
rate

Keywords

  • Blow up
  • Blow-up rate
  • Keller-Segel system
  • Local and global existence

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics

Cite this

Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system. / Kozono, Hideo; Sugiyama, Yoshie.

In: Journal of Evolution Equations, Vol. 8, No. 2, 05.2008, p. 353-378.

Research output: Contribution to journalArticle

@article{3ab07ae6c7fb4b969703866167f2fad0,
title = "Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system",
abstract = "We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u0 ∈ L1 (ℝ2). The local existence time is characterized for u 0 ∈ L1 ∩ Lq* (ℝ2) with 1 < q* < 2. Next, we prove the finite time blow-up of strong solution under the assumption ∥u0∥ L1 > 8π and ∥x|2u0∥L1 < 1/γ·g (∥u0∥L1/8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.",
keywords = "Blow up, Blow-up rate, Keller-Segel system, Local and global existence",
author = "Hideo Kozono and Yoshie Sugiyama",
year = "2008",
month = "5",
doi = "10.1007/s00028-008-0375-6",
language = "English",
volume = "8",
pages = "353--378",
journal = "Journal of Evolution Equations",
issn = "1424-3199",
publisher = "Birkhauser Verlag Basel",
number = "2",

}

TY - JOUR

T1 - Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

AU - Kozono, Hideo

AU - Sugiyama, Yoshie

PY - 2008/5

Y1 - 2008/5

N2 - We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u0 ∈ L1 (ℝ2). The local existence time is characterized for u 0 ∈ L1 ∩ Lq* (ℝ2) with 1 < q* < 2. Next, we prove the finite time blow-up of strong solution under the assumption ∥u0∥ L1 > 8π and ∥x|2u0∥L1 < 1/γ·g (∥u0∥L1/8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

AB - We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u0 ∈ L1 (ℝ2). The local existence time is characterized for u 0 ∈ L1 ∩ Lq* (ℝ2) with 1 < q* < 2. Next, we prove the finite time blow-up of strong solution under the assumption ∥u0∥ L1 > 8π and ∥x|2u0∥L1 < 1/γ·g (∥u0∥L1/8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

KW - Blow up

KW - Blow-up rate

KW - Keller-Segel system

KW - Local and global existence

UR - http://www.scopus.com/inward/record.url?scp=44549087932&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44549087932&partnerID=8YFLogxK

U2 - 10.1007/s00028-008-0375-6

DO - 10.1007/s00028-008-0375-6

M3 - Article

VL - 8

SP - 353

EP - 378

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

SN - 1424-3199

IS - 2

ER -