### Abstract

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation (1) {v_{tt} + v_{t} - v_{xx} + v^{1+σ} = 0, x ∈ R, t > 0, v (0, x) = ε v_{0} (x), v_{t} (0, x) = ε v_{1} (x) in the sub critical case σ ∈ (2 - ε^{3}, 2). We assume that the initial data v_{0}, (1 + ∂_{x})^{-1} v_{1} ∈ L^{∞} ∩ L^{1,a}, a ∈ (0, 1) where L^{1,a} = { ∈ L^{1}; ∥φ∥ _{L1a}, = ∥〈·〉^{a} φ∥ _{L1} < ∞}, 〈x〉 = 1 + x^{2}. Also we suppose that the mean value of initial data ∫_{R} (v_{0} (x) + v_{1} (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^{∞} ∩ L^{1,a}), satisfying the following time decay estimate: ∥v (t)∥ _{L∞} ≤ C ε 〈t〉 ^{-1/σ} for large t > 0, here 2 - ε^{3} < σ < 2.

Original language | English |
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Pages (from-to) | 161-194 |

Number of pages | 34 |

Journal | Journal of Differential Equations |

Volume | 207 |

Issue number | 1 |

DOIs | |

Publication status | Published - 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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## Cite this

*Journal of Differential Equations*,

*207*(1), 161-194. https://doi.org/10.1016/j.jde.2004.06.018