Stability of Timoshenko system coupled with thermal law of Gurtin-Pipkin affecting on shear force

ABSTRACT

In this paper, we consider a Timoshenko type system coupled with parabolic equation that represents the thermal effect given by the Gurtin-Pipkin law while taking into account that the temperature influences on the shear force. We establish the global existence of solution the behavior of the solution. We also prove the lack of the exponential stability when $\xi \ne 0$, we obtain the exponential stability in the case where $\xi =0$ and the parameter p = 1 and also the polynomial stability in the case where $\xi \ne 0$ and $1<p<\frac{3}{2}$ such that p is a parameter related to the thermal memory. One of the novel contribution here is that there is a new number ξ that is different from the number of stability that has been proven previously.

Keywords:

• Timoshenko system
• thermal effect
• energy decay
• asymptotic behavior of solutions
• contraction semigroup
• Lyapunov functional
• Gurtin-Pipkin thermal effect

2010 Mathematics Subject Classifications:

• 35B40
• 74Dxx
• 93D20

Acknowledgments

The authors express sincere thanks to the editors and referees for their constructive comments and suggestions that helped to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Hanni Dridi was supported by the Directorate General for Scientific Research and Technological Development (DGRSDT). Baowei Feng was supported by the National Natural Science Foundation of China [grant number 11701465]. Khaled Zennir thanks Deanship of Scientific Research, Qassim University, for its continuous support.