Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions

Takayoshi Ogawa, Hiroshi Wakui

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

Abstract

We show the non-uniform bound for a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If an initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not remain uniformly bounded in a scaling critical space. In other words, the solution grows up at ∞ in the critical space or blows up in a finite time. Our presenting results correspond to the finite time blowing up result for the two-dimensional case. The proof relies on the logarithmic entropy functional and a generalized version of the Shannon inequality. We also give the sharp constant of the Shannon inequality.

Original languageEnglish
Pages (from-to)145-183
Number of pages39
JournalAnalysis and Applications
Volume14
Issue number1
DOIs
Publication statusPublished - 2016 Jan 1

Keywords

  • Critical space
  • Drift-diffusion
  • Finite time blow up
  • Generalized Shannon's inequality
  • Unbounded solution
  • Virial laws

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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