TY - JOUR
T1 - Decay property of the Timoshenko-Cattaneo system
AU - Mori, Naofumi
AU - Kawashima, Shuichi
N1 - Funding Information:
The first author (N. M.) would like to express his heartiest thanks to Professor Tohru Nakamura at Kyushu University, Professor Yoshihiro Ueda at Kobe University and Professor Hiroshi Takeda at Fukuoka Institute of Technology for giving him valuable advices and helpful comments. The first author was supported by the JSPS Japanese-German Graduate Externship. This work is supported in part by Grant-in-Aid for Scientific Research (A) 22244009.
Publisher Copyright:
© 2016 World Scientific Publishing Company.
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.
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U2 - 10.1142/S0219530515500062
DO - 10.1142/S0219530515500062
M3 - Article
AN - SCOPUS:84928624181
SN - 0219-5305
VL - 14
SP - 393
EP - 413
JO - Analysis and Applications
JF - Analysis and Applications
IS - 3
ER -