Lifespan of solutions to the damped wave equation with a critical nonlinearity

Masahiro Ikeda, Takayoshi Ogawa

Research output: Contribution to journalArticlepeer-review

30 Citations (Scopus)


In the present paper, we study a lifespan of solutions to the Cauchy problem for semilinear damped wave equations(DW) where n≥1, f(u)=±|u|p-1u or |u|p, p≥1, ε>0 is a small parameter, and (u0, u1) is a given initial data. The main purpose of this paper is to prove that if the nonlinear term is f(u)=|u|p and the nonlinear power is the Fujita critical exponent p=pF=1+2n, then the upper estimate to the lifespan is estimated by for all ε∈(0, 1] and suitable data (u0, u1), without any restriction on the spatial dimension. Our proof is based on a test-function method utilized by Zhang [35]. We also prove a sharp lower estimate of the lifespan T(ε) to (DW) in the critical case p=pF.

Original languageEnglish
Pages (from-to)1880-1903
Number of pages24
JournalJournal of Differential Equations
Issue number3
Publication statusPublished - 2016 Aug 5


  • Damped wave equation
  • Fujita exponent
  • Higher dimensions
  • Lifespan
  • Upper bound

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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