Large time behavior and Lp-Lq estimate of solutions of 2-dimensional nonlinear damped wave equations

Takafumi Hosono, Takayoshi Ogawa

Research output: Contribution to journalArticlepeer-review

99 Citations (Scopus)

Abstract

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the Lp-Lq type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the Lp-Lq estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity u αu. Our result covers the whole super critical case α 1, where the α=1 is well known as the Fujita exponent when n=2.

Original languageEnglish
Pages (from-to)82-118
Number of pages37
JournalJournal of Differential Equations
Volume203
Issue number1
DOIs
Publication statusPublished - 2004 Aug 15

Keywords

  • Besov space
  • Cauchy problem
  • Critical exponent
  • Damped wave equation
  • Fourier transform
  • L-L estimate
  • Large time asymptotic behavior
  • Power nonlinearity
  • Self-similar profile
  • Time-global solvability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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