### Abstract

We study the Timoshenko system with Cattaneo's type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal L2 decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko-Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647-667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko-Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier's type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135-157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647-667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko-Cattaneo system.

Original language | English |
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Pages (from-to) | 393-413 |

Number of pages | 21 |

Journal | Analysis and Applications |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 May 1 |

Externally published | Yes |

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### Keywords

- asymptotic behavior
- decay property
- pointwise estimate in the Fourier space
- regularity-loss type
- Timoshenko system with thermal effects

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Analysis and Applications*,

*14*(3), 393-413. https://doi.org/10.1142/S0219530515500062