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Timoshenko system with fractional operator in the memory and spatial fractional thermal effect

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Abstract

We consider the differential system

$$\begin{aligned}&\rho _{1}\varphi _{tt}-\kappa (\varphi _{xx}+\psi _{x})=0,\\&\rho _{2}\psi _{tt}-b\psi _{xx}+\kappa (\varphi _{x}+\psi )-\int _{0}^{\infty }h(s)A^{\sigma }\psi (t-s)ds +\delta \theta _{x}=0,\\&\rho _{3}\theta _{t}+\frac{1}{\beta }\int _{0}^{\infty }g(s)A^{\sigma }\theta (t-s)ds+\delta \psi _{tx}=0, \end{aligned}$$

describing a Timoshenko system with fractional operator in the memory and spatial fractional thermal effect of Gurtin–Pipkin type, which depends on a parameter \(\sigma \in [0,1].\) Under some assumptions on the kernels, the paper proved the global existence of a weak solution. In addition the utilisation of the semigroup method in fractional Hilbert space shows some results about the system stability which is related to the number of stability \( \xi _ {g} \) and the parameter of the fractional order \( \sigma \).

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Acknowledgements

The authors would like to thank the editor and the referees for their very helpful suggestions.

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  1. Laboratory of Applied Mathematics, University of Badji Mokhtar, P.O. Box 12, 23000, Annaba, Algeria

    Hanni Dridi & Abdelhak Djebabla

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  1. Hanni Dridi
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  2. Abdelhak Djebabla
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Correspondence to Hanni Dridi.

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Dridi, H., Djebabla, A. Timoshenko system with fractional operator in the memory and spatial fractional thermal effect. Rend. Circ. Mat. Palermo, II. Ser 70, 593–621 (2021). https://doi.org/10.1007/s12215-020-00513-6

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