Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

Hans Christoph Grunau, Nobuhito Miyake, Shinya Okabe

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

Original languageEnglish
Pages (from-to)353-370
Number of pages18
JournalAdvances in Nonlinear Analysis
Volume10
Issue number1
DOIs
Publication statusPublished - 2021 Jan 1

Keywords

  • biharmonic heat equations
  • global positivity

ASJC Scopus subject areas

  • Analysis

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