Regularity and asymptotic behavior for the keller-segel system of degenerate type with critical nonlinearity

Masashi Mizuno, Takayoshi Ogawa

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We discuss the large time behavior of a weak solution of the Keller-Segel system of degenerate type: { ∂tu- △uα + div(u▽ψ) = 0, t > 0, x ε ℝn,-△ψ + ψ = u, t> 0, x ε ℝn, u(0,x) = u0(x) ≥ 0, x ε ℝn, where α > 1. It is known when the exponent a = 2-2/n then the problem shows the critical situation. In this case, we show that the small data global solution decays and its asymptotic profile converges to the Barenblatt-Pattle solution U(t) - (1 + t) -n/σ(A- |x|2/(1 + t)1/(σ-1)^ in L1 such as ||u(t)-u(t)||1 ≤C(1 + t)-v, where v > 0 is depending on n and the regularity of the solution. To show this, we employ the forward self-similar transform and use the entropy dissipation term to derive the asymptotic profile due to Carrillo-Toscani [12] and Ogawa [47]. The Hölder continuity of the weak solution for the forward self-similar equation plays a crucial role. We derive the uniform Hölder continuity by using the rescaled alternative selection originated by DiBenedetto-Friedman [18, 19].

Original languageEnglish
Pages (from-to)375-433
Number of pages59
JournalJournal of Mathematical Sciences (Japan)
Volume20
Issue number3
Publication statusPublished - 2013

Keywords

  • Degenerate keller-segel system
  • Hölder regularity
  • Uniform asymptotic estimates

ASJC Scopus subject areas

  • Mathematics(all)

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