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The Long-Time Stability of Solutions for Intermittently Controlled Viscoelastic Wave Equations with Memory Terms

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Abstract

In this paper, we investigate the asymptotic stability of global solutions to the following intermittently controlled semilinear viscoelastic equation with memory term

$$\begin{aligned}&u_{tt}-\varDelta u_{tt}-\varDelta u+\int _{0}^{\tau }k(s)\varDelta u(t-s)ds+h_{1}(t)u_{t} -h_{2}(t)\varDelta u_{t}\\&\quad =f(u),\;(x,t)\in \varOmega \times [0,\infty ), \end{aligned}$$

under the null Dirichlet boundary condition and \(\tau \in \{t,\infty \}\). By virtue of appropriate new Lyapunov functional and Łojasiewicz–Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integral positive and positive-negative, respectively. Moreover, under the assumptions of on–off or sign-changing dampings, we derive the asymptotic stability of solutions.

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Acknowledgements

The authors would like to deeply thank all the reviewers for their insightful and constructive comments. This work is supported by the Natural Science Foundation of Shandong Province of China (No. ZR2019MA072) and the Fundamental Research Funds for the Central Universities (Nos. 201861002, 201964008)

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  1. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, People’s Republic of China

    Zhiqing Liu & Zhong Bo Fang

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  1. Zhiqing Liu
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Correspondence to Zhong Bo Fang.

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Liu, Z., Fang, Z.B. The Long-Time Stability of Solutions for Intermittently Controlled Viscoelastic Wave Equations with Memory Terms. Appl Math Optim 83, 1991–2016 (2021). https://doi.org/10.1007/s00245-019-09616-8

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