### Abstract

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

Original language | English |
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Pages (from-to) | 1165-1185 |

Number of pages | 21 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 Mar |

### Keywords

- Damped wave equation
- Large time asymptotics

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*358*(3), 1165-1185. https://doi.org/10.1090/S0002-9947-05-03818-3