The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

Hiroyuki Takamura, Kyouhei Wakasa

Research output: Contribution to journalArticle

36 Citations (Scopus)

Abstract

The final open part of Strauss' conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang (2006) [18], or Zhou (2007) [21] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers.In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou (1995) [10], the lifespan T(ε) of solutions of utt-δu=u2 in R4×[0,∞) with the initial data u(x,0)=εf(x),ut(x,0)=εg(x) of a small parameter ε>0, compactly supported smooth functions f and g, has an estimate. exp(cε-2)≤T(ε)≤exp(Cε-2), where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations.

Original languageEnglish
Pages (from-to)1157-1171
Number of pages15
JournalJournal of Differential Equations
Volume251
Issue number4-5
DOIs
Publication statusPublished - 2011 Aug 15
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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