### Abstract

As a continued work of [18], we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of regularity-loss. Firstly, with the modification of a priori estimates in [18], we construct global solutions to the Timoshenko system pertaining to data in the Besov space with the regularity s= 3/2. Owing to the weaker dissipative mechanism, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions, so it is almost impossible to obtain the optimal decay rates in the critical space. To overcome the outstanding difficulty, we develop a new frequency-localization time-decay inequality, which captures the information related to the integrability at the high-frequency part. Furthermore, by the energy approach in terms of high-frequency and low-frequency decomposition, we show the optimal decay rate for Timoshenko system in critical Besov spaces, which improves previous works greatly.

Original language | English |
---|---|

Pages (from-to) | 5533-5553 |

Number of pages | 21 |

Journal | Journal of Differential Equations |

Volume | 259 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Critical Besov spaces
- Global existence
- Minimal decay regularity
- Timoshenko system

### ASJC Scopus subject areas

- Analysis

### Cite this

**Global existence and minimal decay regularity for the Timoshenko system : The case of non-equal wave speeds.** / Xu, Jiang; Mori, Naofumi; Kawashima, Shuichi.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 259, no. 11, pp. 5533-5553. https://doi.org/10.1016/j.jde.2015.06.041

}

TY - JOUR

T1 - Global existence and minimal decay regularity for the Timoshenko system

T2 - The case of non-equal wave speeds

AU - Xu, Jiang

AU - Mori, Naofumi

AU - Kawashima, Shuichi

PY - 2015/1/1

Y1 - 2015/1/1

N2 - As a continued work of [18], we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of regularity-loss. Firstly, with the modification of a priori estimates in [18], we construct global solutions to the Timoshenko system pertaining to data in the Besov space with the regularity s= 3/2. Owing to the weaker dissipative mechanism, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions, so it is almost impossible to obtain the optimal decay rates in the critical space. To overcome the outstanding difficulty, we develop a new frequency-localization time-decay inequality, which captures the information related to the integrability at the high-frequency part. Furthermore, by the energy approach in terms of high-frequency and low-frequency decomposition, we show the optimal decay rate for Timoshenko system in critical Besov spaces, which improves previous works greatly.

AB - As a continued work of [18], we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of regularity-loss. Firstly, with the modification of a priori estimates in [18], we construct global solutions to the Timoshenko system pertaining to data in the Besov space with the regularity s= 3/2. Owing to the weaker dissipative mechanism, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions, so it is almost impossible to obtain the optimal decay rates in the critical space. To overcome the outstanding difficulty, we develop a new frequency-localization time-decay inequality, which captures the information related to the integrability at the high-frequency part. Furthermore, by the energy approach in terms of high-frequency and low-frequency decomposition, we show the optimal decay rate for Timoshenko system in critical Besov spaces, which improves previous works greatly.

KW - Critical Besov spaces

KW - Global existence

KW - Minimal decay regularity

KW - Timoshenko system

UR - http://www.scopus.com/inward/record.url?scp=84941801047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941801047&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2015.06.041

DO - 10.1016/j.jde.2015.06.041

M3 - Article

AN - SCOPUS:84941801047

VL - 259

SP - 5533

EP - 5553

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 11

ER -