Almost global solutions of semilinear wave equations with the critical exponent in high dimensions in memory of Professor Rentaro Agemi

Hiroyuki Takamura, Kyouhei Wakasa

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

We are interested in the "almost" global-in-time existence of classical solutions in the general theory for nonlinear wave equations. All the three such cases are known to be sharp due to blow-up results in the critical case for model equations. However, it is known that we have a possibility to get the global-in-time existence for two of them in low space dimensions if the nonlinear term is of derivatives of the unknown function and satisfies the so-called null condition, or non-positive condition. But another one for the quadratic term in four space dimensions is out of the case as the nonlinear term should include a square of the unknown function itself. In this paper, we get one more example guaranteeing the sharpness of the almost global-in-time existence in four space dimensions. It is also the first example of the blow-up of classical solutions for non-single and indefinitely signed term in high dimensions. Such a term arises from the neglect of derivative-loss factors in Duhamel's formula for positive and single nonlinear term. This fact may help us to describe a criterion to get the global-in-time existence in this critical situation.

Original languageEnglish
Pages (from-to)187-229
Number of pages43
JournalNonlinear Analysis, Theory, Methods and Applications
Volume109
DOIs
Publication statusPublished - 2014 Nov
Externally publishedYes

Keywords

  • Critical exponent
  • High dimensions
  • Lifespan
  • Semilinear wave equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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