Damped wave equation in the subcritical case

  • Hayashi N
  • Kaikina E
  • Naumkin P
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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

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