Longtime Dynamics of a Semilinear Lamé System

Abstract

This paper is concerned with longtime dynamics of semilinear Lamé systems

$$\begin{aligned} \partial ^2_t u - \mu \Delta u - (\lambda + \mu ) \nabla \mathrm{div} u + \alpha \partial _t u + f(u) = b, \end{aligned}$$

defined in bounded domains of \({\mathbb {R}}^3\) with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to a critical forcing f(u). Writing \(\lambda + \mu \) as a positive parameter \(\varepsilon \), we discuss some physical aspects of the limit case \(\varepsilon \rightarrow 0\). Then, we show the upper-semicontinuity of attractors with respect to the parameter when \(\varepsilon \rightarrow 0\). To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before.

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References

  1. 1.

    Achenbach, J.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)

    Google Scholar 

  2. 2.

    Alabau, F., Komornik, V.: Boundary observability, controllability, and stabilization of linear elastodynamic systems. SIAM J. Control Optim. 37, 521–542 (1999)

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Astaburuaga, M.A., Charão, R.C.: Stabilization of the total energy for a system of elasticity with localized dissipation. Differ. Integral Equ. 15, 1357–1376 (2002)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bchatnia, A., Guesmia, A.: Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain. Math. Control Relat. Fields 4, 451–463 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Benaissa, A., Gaouar, S.: Asymptotic stability for the Lamé system with fractional boundary damping. Comput. Math. Appl. 77, 1331–1346 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Belishev, M.I., Lasiecka, I.: The dynamical Lame system: regularity of solutions, boundary controllability and boundary data continuation. ESAIM: Control Optim. Calc. Var. 18, 143–167 (2002)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Cavalcanti, M.M., Fatori, L.H., Ma, T.F.: Attractors for wave equations with degenerate memory. J. Differ. Equ. 260, 56–83 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cerveny, V.: Seismic Ray Theory. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  10. 10.

    Chueshov, I.: Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham (2015)

    Google Scholar 

  11. 11.

    Chueshov, I., Lasiecka, I.: Von Karman evolution equations: well-posedness and long-time dynamics. In: Springer Monographs in Mathematics. Springer, New York (2010)

  12. 12.

    Ciarlet, P.G.: Mathematical Elasticity, Three-Dimensional Elasticity, vol. I. North-Holland, Amsterdam (1988)

    Google Scholar 

  13. 13.

    Geredeli, P.G., Lasiecka, I.: Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity. Nonlinear Anal. Theory Methods Appl. 91, 72–92 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Goodway, B.: AVO and Lamé constants for rock parameterization and fluid detection. CSEG Rec. 26, 30–60 (2001)

    Google Scholar 

  15. 15.

    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (2010)

    Google Scholar 

  16. 16.

    Hale, J.K., Raugel, G.: Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. J. Differ. Equ. 73, 197–214 (1988)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Horn, M.A.: Stabilization of the dynamic system of elasticity by nonlinear boundary feedback. In: Hoffmann, K.-H., Leugering, G., Troltzsch, F. (eds.) Optimal Control of Partial Differential Equations, International Conference in Chemnitz, Germany, April 20–25, 1998. Springer, Basel (1999)

    Google Scholar 

  18. 18.

    Hudson, J.: The Excitation and Propagation of Elastic Waves. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  19. 19.

    Ji, S., Sun, S., Wang, Q., Marcotte, D.: Lamé parameters of common rocks in the Earth’s crust and upper mantle. J. Geophys. Res. 115 (2010) article B06314

  20. 20.

    Kline, M., Kay, I.: Electromagnetic Theory and Geometrical Optics. Interscience, New York (1965)

    Google Scholar 

  21. 21.

    Lagnese, J.: Boundary stabilization of linear elastodynamic systems. SIAM J. Control Optim 21, 968–984 (1983)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Lakes, R., Wojciechowski, K.W.: Negative compressibility, negative Poisson’s ratio, and stability. Phys. Stat. Sol. (B) 245, 545–551 (2008)

    Article  Google Scholar 

  23. 23.

    Lions, J.L.: Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1. Masson, Paris (1988)

    Google Scholar 

  24. 24.

    Liu, W.-J., Krstić, M.: Strong stabilization of the system of linear elasticity by a Dirichlet boundary feedback. IMA J. Appl. Math. 65, 109–121 (2000)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Love, A. E. H.: A Treatise on Mathematical Theory of Elasticity. Cambridge (1892)

  26. 26.

    Ma, T.F., Monteiro, R.N.: Singular limit and long-time dynamics of Bresse systems. SIAM J. Math. Anal. 49(4), 2468–2495 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Moore, B., Jaglinski, T., Stone, D.S., Lakes, R.S.: Negative incremental bulk modulus in foams. Philos. Mag. Lett. 86, 651–659 (2006)

    Article  Google Scholar 

  28. 28.

    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44. Springer, Berlin (2012)

    Google Scholar 

  29. 29.

    Poisson, S.D.: Mémoire sur l’équilibre et le mouvement des corps élastiques. Mémoires de l’Académie Royal des Sciences de l’Institut de France VII(I), 357–570 (1829)

  30. 30.

    Pujol, J.: Elastic Wave Propagation and Generation in Seismology. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  31. 31.

    Simon, J.: Compact sets in the space \(L^p (O, T; B)\). Annali di Matematica 146, 65–96 (1986)

    Article  Google Scholar 

  32. 32.

    Teodorescu, P.P.: Treatise on Classical Elasticity, Theory and Related Problems. Springer, Dordrecht (2013)

    Google Scholar 

  33. 33.

    Timoshenko, S.P.: History of the Strength of Materials. McGraw-Hill, New York (1953)

    Google Scholar 

  34. 34.

    Yamamoto, K.: Exponential energy decay of solutions of elastic wave equations with the Dirichlet condition. Math. Scand. 65, 206–220 (1989)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors thank the anonymous referee for his/her comments and remarks that improved the previous version of this paper.

Funding

L. E. Bocanegra-Rodríguez was supported by CAPES, finance code 001 (Ph.D. Scholarship). M. A. Jorge Silva was partially supported by Fundação Araucária Grant 066/2019 and CNPq Grant 301116/2019-9. T. F. Ma was partially supported by CNPq Grant 312529/2018-0 and FAPESP Grant 2019/11824-0. P. N. Seminario-Huertas was fully supported by INCTMat-CAPES Grant 88887.507829/2020-00.

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Correspondence to Paulo Nicanor Seminario-Huertas.

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Bocanegra-Rodríguez, L.E., Silva, M.A.J., Ma, T.F. et al. Longtime Dynamics of a Semilinear Lamé System. J Dyn Diff Equat (2021). https://doi.org/10.1007/s10884-021-09955-7

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Keywords

  • System of elasticity
  • Global attractor
  • Gradient system
  • Upper-semicontinuity

Mathematics Subject Classification

  • 35B41
  • 74H40
  • 74B05