Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space

Hideo Kozono, Yoshie Sugiyama, Takuya Wachi

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We prove the existence and uniqueness of solutions (u,v) to the Keller-Segel system of parabolic-parabolic type in Rn for n≥. 3 in the scaling invariant class u∈Lq(0,T;Lr(Rn)), v∈Lr~(0,T;Hβ,r~(Rn)), where 2/. q+. n/. r= 2, 2/q~+n/r~=2β provided the initial data (u0,v0) is chosen as u0∈Ln/2(Rn), v0∈H2α,n/2α(Rn) for n/2(n+. 2) < α ≤ 1/2. In particular, our uniqueness result holds for all n≥. 2 even though we impose an assumption only on u such as Lq(0,T;Lr(Rn)) for 2/. q+. n/. r= 2 with n/2 < r. As for the marginal case when r= n/2, we show that if n≥. 3 and if u∈C([0,T);Ln/2(Rn)), ∇;v∈C([0,T);Ln(Rn)), then (u,v) is the only solution.

Original languageEnglish
Pages (from-to)1213-1228
Number of pages16
JournalJournal of Differential Equations
Volume252
Issue number2
DOIs
Publication statusPublished - Jan 15 2012

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Existence and Uniqueness Theorem
Mild Solution
Scaling
Invariant
Existence and Uniqueness of Solutions
Uniqueness
Class

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space. / Kozono, Hideo; Sugiyama, Yoshie; Wachi, Takuya.

In: Journal of Differential Equations, Vol. 252, No. 2, 15.01.2012, p. 1213-1228.

Research output: Contribution to journalArticle

Kozono, Hideo ; Sugiyama, Yoshie ; Wachi, Takuya. / Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space. In: Journal of Differential Equations. 2012 ; Vol. 252, No. 2. pp. 1213-1228.
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