Damped wave equation with a critical nonlinearity

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

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity {∂t2u+∂tu-△u+λu 1+2/n = 0, x ∈ Rn, t>0, u(0,x) = εu 0 (x), ∂tu(0,x) = εu1 (x), x ∈ Rn, where ε > 0, and space dimensions n = 1,2,3, Assume that the initial data u0 ∈ Hδ,0 ∩ H 0,δ, u1 ∈ Hδ-1,0 ∩ H -1,δ, where δ>n/2, weighted Sobolev spaces are H l,m = {Φ ∈ L2;∥xmi∂x l Φ (x)∥L2 <∞}, 〈x〉 = √1+x2. Also we suppose that λθ 2/n>0∫u0(x)dx>0, where θ = ∫ (u 0 (x) + u1 (x)) dx. Then we prove that there exists a positive ε0 such that the Cauchy problem above has a unique global solution u ∈ C ([0, ∞); Hδ,0) satisfying the time decay property ∥u(t)-εθG(t,x)e-ℓ(t)∥ Lp ≤Cε1+2/ng-1-n/2(t) 〈t〉-n/2(1-1/p) for all t > 0, 1 ≤ p ≤ ∞ where ε ∈ (0, ε0].

Original languageEnglish
Pages (from-to)1165-1185
Number of pages21
JournalTransactions of the American Mathematical Society
Issue number3
Publication statusPublished - 2006 Mar


  • Damped wave equation
  • Large time asymptotics


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