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

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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Corresponding author

Correspondence to Wenjun Liu.

Additional information

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).

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Cite this article

Liu, W., Chen, D. & Chen, Z. Long-Time Behavior for a Thermoelastic Microbeam Problem with Time Delay and the Coleman-Gurtin Thermal Law. Acta Math Sci 41, 609–632 (2021). https://doi.org/10.1007/s10473-021-0220-3

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Key words

  • Microbeam problem
  • time delay
  • Coleman-Gurtin law
  • global attractor
  • exponential attractor

2010 MR Subject Classification

  • 35B41
  • 35L76
  • 74F05
  • 74H40