### 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 |
---|---|

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 |

### Fingerprint

### 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

**Decay property of the Timoshenko-Cattaneo system.** / Mori, Naofumi; Kawashima, Shuichi.

Research output: Contribution to journal › Article

*Analysis and Applications*, vol. 14, no. 3, pp. 393-413. https://doi.org/10.1142/S0219530515500062

}

TY - JOUR

T1 - Decay property of the Timoshenko-Cattaneo system

AU - Mori, Naofumi

AU - Kawashima, Shuichi

PY - 2016/5/1

Y1 - 2016/5/1

N2 - 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.

AB - 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.

KW - asymptotic behavior

KW - decay property

KW - pointwise estimate in the Fourier space

KW - regularity-loss type

KW - Timoshenko system with thermal effects

UR - http://www.scopus.com/inward/record.url?scp=84928624181&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928624181&partnerID=8YFLogxK

U2 - 10.1142/S0219530515500062

DO - 10.1142/S0219530515500062

M3 - Article

AN - SCOPUS:84928624181

VL - 14

SP - 393

EP - 413

JO - Analysis and Applications

JF - Analysis and Applications

SN - 0219-5305

IS - 3

ER -