Finite time blow up for a solution to system of the drift–diffusion equations in higher dimensions

Masaki Kurokiba, Takayoshi Ogawa

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

We discuss the existence of a blow-up solution for a multi-component parabolic–elliptic drift–diffusion model in higher space dimensions. We show that the local existence, uniqueness and well-posedness of a solution in the weighted L2 spaces. Moreover we prove that if the initial data satisfies certain conditions, then the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift–diffusion equation proved by Nagai (J Inequal Appl 6:37–55, 2001) and Nagai et al. (Hiroshima J Math 30:463–497, 2000) and gravitational interaction of particles by Biler (Colloq Math 68:229–239, 1995), Biler and Nadzieja (Colloq Math 66:319–334, 1994, Adv Differ Equ 3:177–197, 1998). We generalize the result in Kurokiba and Ogawa (Differ Integral Equ 16:427–452, 2003, Differ Integral Equ 28:441–472, 2015) and Kurokiba (Differ Integral Equ 27(5–6):425–446, 2014) for the multi-component problem and give a sufficient condition for the finite time blow up of the solution. The condition is different from the one obtained by Corrias et al. (Milan J Math 72:1–28, 2004).

Original languageEnglish
Pages (from-to)231-253
Number of pages23
JournalMathematische Zeitschrift
Volume284
Issue number1-2
DOIs
Publication statusPublished - 2016 Oct 1

Keywords

  • 35K55
  • 35Q60
  • Primary 35K15
  • Secondary 78A35

ASJC Scopus subject areas

  • Mathematics(all)

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