### Abstract

We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

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

Pages (from-to) | 1494-1518 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 258 |

Issue number | 5 |

DOIs | |

Publication status | Published - Mar 5 2015 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Differential Equations*,

*258*(5), 1494-1518. https://doi.org/10.1016/j.jde.2014.11.003

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

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 258, no. 5, pp. 1494-1518. https://doi.org/10.1016/j.jde.2014.11.003

}

TY - JOUR

T1 - Global existence and optimal decay rates for the Timoshenko system

T2 - The case of equal wave speeds

AU - Mori, Naofumi

AU - Xu, Jiang

AU - Kawashima, Shuichi

PY - 2015/3/5

Y1 - 2015/3/5

N2 - We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

AB - We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

UR - http://www.scopus.com/inward/record.url?scp=84920684659&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920684659&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2014.11.003

DO - 10.1016/j.jde.2014.11.003

M3 - Article

AN - SCOPUS:84920684659

VL - 258

SP - 1494

EP - 1518

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 5

ER -