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Long-Time Behavior for a Thermoelastic Microbeam Problem with Time Delay and the Coleman-Gurtin Thermal Law

Authors
  • Liu, Wenjun1
  • Chen, Dongqin1
  • Chen, Zhijing1
  • 1 Nanjing University of Information Science and Technology, Nanjing, 210044, China , Nanjing (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jan 29, 2021
Volume
41
Issue
2
Pages
609–632
Identifiers
DOI: 10.1007/s10473-021-0220-3
Source
Springer Nature
Keywords
License
Yellow

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.

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