Damped wave equation in the subcritical case

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation (1) {vtt + vt - vxx + v1+σ = 0, x ∈ R, t > 0, v (0, x) = ε v0 (x), vt (0, x) = ε v1 (x) in the sub critical case σ ∈ (2 - ε3, 2). We assume that the initial data v0, (1 + ∂x)-1 v1 ∈ L ∩ L1,a, a ∈ (0, 1) where L1,a = { ∈ L1; ∥φ∥ L1a, = ∥〈·〉a φ∥ L1 < ∞}, 〈x〉 = 1 + x2. Also we suppose that the mean value of initial data ∫R (v0 (x) + v1 (x)) dx > 0. Then there exists a positive value ε such that the Cauchy problem (1) has a unique global solution v (t, x) ∈ C ([0, ∞); L ∩ L1,a), satisfying the following time decay estimate: ∥v (t)∥ L∞ ≤ C ε 〈t〉 -1/σ for large t > 0, here 2 - ε3 < σ < 2.

Original languageEnglish
Pages (from-to)161-194
Number of pages34
JournalJournal of Differential Equations
Volume207
Issue number1
DOIs
Publication statusPublished - 2004 Dec 1

Keywords

  • Asymptotic expansion
  • Damped wave equation
  • Large time behavior
  • Subcritical nonlinearity

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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