Damped wave equation in the subcritical case

  • Hayashi N
  • Kaikina E
  • Naumkin P
  • 3

    Readers

    Mendeley users who have this article in their library.
  • 26

    Citations

    Citations of this article.

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)-1v1∈ L∞∩ L1,a, a ∈ (0, 1) where L1,a= { ∈ L1; ∥φ∥L1a, = ∥〈·〉aφ∥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/σfor large t > 0, here 2 - ε3< σ < 2. © 2004 Elsevier Inc. All rights reserved.

Author-supplied keywords

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

Get free article suggestions today

Mendeley saves you time finding and organizing research

Sign up here
Already have an account ?Sign in

Find this document

Authors

Cite this document

Choose a citation style from the tabs below

Save time finding and organizing research with Mendeley

Sign up for free