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Global Attractor for a Coupled Wave and Plate Equation with Nonlocal Weak Damping on Riemannian Manifolds

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Abstract

In this paper, we consider the long time behavior for a coupled system on Riemannian manifold consisting of the plate equation and the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity. We obtain the existence of the global attractor for the coupled system by semi-group method and multiplier method.

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References

  1. Hale, J.K.: Functional differential equations. In: Analytic Theory of Differential Equations, pp. 9–22 (1971). Springer, Berlin. https://doi.org/10.1007/BFb0060406

  2. Belleri, V., Pata, V.: Attractors for semilinear strongly damped wave equations on \(\mathbb{R} ^{3}\). Discrete Contin. Dyn. Syst. 7(4), 719 (2001). https://doi.org/10.3934/dcds.2001.7.719

    Article  MATH  Google Scholar 

  3. Ghidaglia, J.M., Marzocchi, A.: Longtime behaviour of strongly damped wave equations, global attractors and their dimension. SIAM J. Math. Anal. 22(4), 879–895 (1991). https://doi.org/10.1137/0522057

    Article  MathSciNet  MATH  Google Scholar 

  4. Arrieta, J., Carvalho, A.N., Hale, J.K.: A damped hyperbolic equation with critical exponent. Commun. Partial Differ. Equ. 17(5–6), 841–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ball, J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10(1 &2), 31–52 (2004). https://doi.org/10.3934/dcds.2004.10.31

    Article  MathSciNet  MATH  Google Scholar 

  6. Yang, Z.J., Ding, P.Y., Li, L.: Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity. J. Math. Anal. Appl. 42(2), 485–510 (2016). https://doi.org/10.1016/j.jmaa.2016.04.079

    Article  MathSciNet  MATH  Google Scholar 

  7. Yang, Z.J., Liu, Z.M., Niu, P.P.: Exponential attractor for the wave equation with structural damping and supercritical exponent. Commun. Contemp. Math. 18(06), 1550055 (2016). https://doi.org/10.1142/S0219199715500558

    Article  MathSciNet  MATH  Google Scholar 

  8. Feireisl, E.: Attractors for wave equations with nonlinear dissipation and critical exponent. Compt. rendus del’Académie des sciences. Série 1 Mathématique 315(5), 551–555 (1992)

  9. Chueshov, I., Lasiecka, I.: Long-time behavior of second order evolution equations with nonlinear damping. Mem. Am. Math. Soc. (2008). https://doi.org/10.1090/memo/0912

    Article  MathSciNet  MATH  Google Scholar 

  10. Chueshov, I.: Long-time dynamics of Kirchhoff wave models with strong nonlinear damping. J. Differ. Equ. 252(2), 1229–1262 (2012). https://doi.org/10.1016/j.jde.2011.08.022

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding, P.Y., Yang, Z.J., Li, Y.N.: Global attractor of the Kirchhoff wave models with strong nonlinear damping. Appl. Math. Lett. 76, 40–45 (2018). https://doi.org/10.1016/j.aml.2017.07.008

    Article  MathSciNet  MATH  Google Scholar 

  12. Cavalcanti, M., Silva, M.A.J., Webler, C.: Exponential stability for the wave equation with degenerate nonlocal weak damping. Isr. J. Math. 219(1), 189–213 (2017). https://doi.org/10.1007/s11856-017-1478-y

    Article  MathSciNet  MATH  Google Scholar 

  13. Chueshov, I.: Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping. J. Abstr. Differ. Equ. Appl. 1, 86–106 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Silva, M.A.J., Narciso, V.: Attractors and their properties for a class of nonlocal extensible beams. Discrete Contin. Dyn. Syst. 35(3), 985 (2015). https://doi.org/10.3934/dcds.2015.35.985

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhao, C.Y., Zhao, C.X., Zhong, C.K.: Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping. J. Math. Anal. Appl. 490(1), 124186 (2020). https://doi.org/10.1016/j.jmaa.2020.124186

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhao, C.Y., Zhong, C.K., Tang, Z.J.: Asymptotic behaviour of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity. arXiv preprint (2021). Arxiv:2108.07395

  17. Wang, J.C., Yao, P.F.: On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Commun. Pure Appl. Anal. 21(6), 1857–1871 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rodrigues, J.H., Madhumita, R.: Existence of global attractors for a semilinear wave equation with nonlinear boundary dissipation and nonlinear interior and boundary sources with critical exponents. Appl. Math. Optim. 86(3), 35 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lasiecka, I., Rodrigues, J.H.: Weak and strong semigroups in structural acoustic Kirchhoff–Boussinesq interactions with boundary feedback. J. Differ. Equ. 298, 387–429 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bucci, F., Chueshov, I., Lasiecka, I.: Global attractor for a composite system of nonlinear wave and plate equations. Commun. Pure Appl. Anal. 6(1), 113 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bucci, F., Chueshov, I.: Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. arXiv preprint (2008). arXiv:0806.4550

  22. Aliev, A.B., Farhadova, Y.M.: Existence of global attractors for the coupled system of suspension bridge equations. Azerbaijan J. Math. 11(2), 105–124 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Qin, Y.M.: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, vol. 184. Springer, Berlin (2008)

  24. Bucci, F., Toundykov, D.: Finite-dimensional attractor for a composite system of wave/plate equations with localized damping. Nonlinearity 23(9), 2271–2306 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lax, P.: Functional Analysis. Wiley, New York (2002)

    MATH  Google Scholar 

  26. Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. 1. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  27. Yao, P.F.: Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach. CRC Press, Boca Raton (2011). https://doi.org/10.1201/b11042

  28. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (1976)

    Book  MATH  Google Scholar 

  29. Chueshov, I., Eller, M., Lasiecka, I.: On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun. Partial Differ. Equ. 27, 1901–1951 (2002). https://doi.org/10.1081/PDE-120016132

    Article  MathSciNet  MATH  Google Scholar 

  30. Showalter, R.E.: Monotone Operators and Nonlinear Partial Differential Equations in Banach Space. AMS Mathematical Surveys and Monographs, Providence (1997)

    MATH  Google Scholar 

  31. Chueshov, I.: Dynamics of Quasi-stable Dissipative Systems, vol. 18. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22903-4

  32. Simon, J.: Compact sets in the space \(l^p([0, t]; b)\). Ann. Mat. Pura Appl. 146(1), 65–96 (1986). https://doi.org/10.1007/BF01762360

    Article  MathSciNet  MATH  Google Scholar 

  33. Chueshov, I., Lasiecka, I.: Attractors for second-order evolution equations with a nonlinear damping. J. Dyn. Differ. Equ. 16, 469–512 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by National Natural Science Foundation of China (Grant No. 61473126).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Qingqing Peng & Zhifei Zhang

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Qingqing Peng & Zhifei Zhang

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  1. Qingqing Peng
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Correspondence to Zhifei Zhang.

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Peng, Q., Zhang, Z. Global Attractor for a Coupled Wave and Plate Equation with Nonlocal Weak Damping on Riemannian Manifolds. Appl Math Optim 88, 28 (2023). https://doi.org/10.1007/s00245-023-09998-w

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