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

Hideo Kozono; Yoshie Sugiyama

doi:10.1007/s00028-008-0375-6

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

Hideo Kozono^*, Yoshie Sugiyama

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

42 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 ∩ L^q* (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.

Original language	English
Pages (from-to)	353-378
Number of pages	26
Journal	Journal of Evolution Equations
Volume	8
Issue number	2
DOIs	https://doi.org/10.1007/s00028-008-0375-6
Publication status	Published - 2008 May
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.1007/s00028-008-0375-6

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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.",

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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.

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

Abstract

Keywords

ASJC Scopus subject areas

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