LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW

doi:10.1007/s10473-021-0220-3

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (2): 609-632.doi: 10.1007/s10473-021-0220-3

LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW

刘文军, 陈冬琴, 陈志婧

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

收稿日期:2020-03-17 出版日期:2021-04-25 发布日期:2021-04-29
通讯作者: Wenjun LIU E-mail:wjliu@nuist.edu.cn
作者简介:Dongqin CHEN,E-mail:cdqlinsey@163.com;Zhijing CHEN,E-mail:zhijingchen@139.com
基金资助:
This work was supported by the National Natural Science Foundation of China (11771216 and 11901306), the Key Research and Development Program of Jiangsu Province (Social Development) (BE2019725), and the Natural Science Foundation of Jiangsu Province (SBK2017043142).

LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW

Wenjun LIU, Dongqin CHEN, Zhijing CHEN

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received:2020-03-17 Online:2021-04-25 Published:2021-04-29
Contact: Wenjun LIU E-mail:wjliu@nuist.edu.cn
About author:Dongqin CHEN,E-mail:cdqlinsey@163.com;Zhijing CHEN,E-mail:zhijingchen@139.com
Supported by:
This work was supported by the National Natural Science Foundation of China (11771216 and 11901306), the Key Research and Development Program of Jiangsu Province (Social Development) (BE2019725), and the Natural Science Foundation of Jiangsu Province (SBK2017043142).

摘要/Abstract

摘要： This study addresses long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, the convolution kernel of which entails an extremely weak dissipation in the thermal law. By using the semigroup theory, we first establish the existence of global weak and strong solutions as well as their continuous dependence on the initial data in appropriate function spaces, under suitable assumptions on the weight of time delay term, the external force term and the nonlinear term. We then prove that the system is quasi-stable and has a gradient on bounded variant sets, and obtain the existence of a global attractor whose fractal dimension is finite. A result on the exponential attractor of the system is also proved.

关键词: Microbeam problem, time delay, Coleman-Gurtin law, global attractor, exponential attractor

Abstract: This study addresses long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, the convolution kernel of which entails an extremely weak dissipation in the thermal law. By using the semigroup theory, we first establish the existence of global weak and strong solutions as well as their continuous dependence on the initial data in appropriate function spaces, under suitable assumptions on the weight of time delay term, the external force term and the nonlinear term. We then prove that the system is quasi-stable and has a gradient on bounded variant sets, and obtain the existence of a global attractor whose fractal dimension is finite. A result on the exponential attractor of the system is also proved.

Key words: Microbeam problem, time delay, Coleman-Gurtin law, global attractor, exponential attractor

中图分类号:

35B41

刘文军, 陈冬琴, 陈志婧. LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW[J]. 数学物理学报(英文版), 2021, 41(2): 609-632.

Wenjun LIU, Dongqin CHEN, Zhijing CHEN. LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW[J]. Acta mathematica scientia,Series B, 2021, 41(2): 609-632.

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LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW

LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW

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