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Large-time behavior of solutions to degenerate damped hyperbolic equations

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Abstract

We investigate the asymptotic behavior of solutions to damped hyperbolic equations involving strongly degenerate differential operators. First we establish the existence of a global attractor for the damped hyperbolic equation under consideration. Then we prove the finite dimensionality of the global attractor.

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References

  1. Babin A. V. and Vishik M. I., “Regular attractors of semigroups and evolution equations,” J. Math. Pures Appl., 62, No. 4, 441–491 (1983).

    MathSciNet  MATH  Google Scholar 

  2. Chepyzhov V. V. and Vishik M. I., Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence (2002) (Amer. Math. Soc. Colloq. Publ.; V. 49).

    MATH  Google Scholar 

  3. Robinson J. C., Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge (2001).

    Book  MATH  Google Scholar 

  4. Temam R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York (1988).

    Book  MATH  Google Scholar 

  5. Grushin V. V., “On a class of hypoelliptic operators,” Math. USSR-Sb., 12, No. 3, 458–476 (1970).

    Article  MATH  Google Scholar 

  6. Thuy N. T. C. and Tri N. M., “Some existence and nonexistence results for boundary value problems (BVP) for semilinear elliptic degenerate operators,” Russ. J. Math. Phys., 9, No. 3, 366–371 (2002).

    MathSciNet  MATH  Google Scholar 

  7. Thuy P. T. and Tri N. M., “Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations,” Nonlinear Diff. Equ. Appl. (NoDEA), 19, 279–298 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. Kogoj A. E. and Lanconelli E., “On semilinear p-Laplace equation,” Nonlinear Anal., 75, 4637–4649 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  9. Luyen D. T. and Tri N. M., “Existence of solutions to boundary value problems for semilinear differential equations,” Math. Notes, 97, No. 1, 73–84 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. Tri N. M., Semilinear Degenerate Elliptic Differential Equations. Local and Global Theories, Lambert Acad. Publ., Saarbrücken (2010).

    Google Scholar 

  11. Tri N. M., Recent Progress in the Theory of Semilinear Equations Involving Degenerate Elliptic Differential Operators, Publ. House Sci. Technology., Hanoi (2014).

    Google Scholar 

  12. Hale J. K., “Asymptotic behavior and dynamics in infinite dimensions,” in: Nonlinear Differential Equations (Granada, 1984), Pitman, Boston, MA, 1985, pp. 1–42 (Res. Notes Math.; V. 132).

    Google Scholar 

  13. Hale J. K. and Raugel G., “Attractors for dissipative evolutionary equations,” in: Proc. Conf. EQUADIFF-91 (Univ. Barcelona, Barcelona, August 26–31, 1991) (C.Perelló, C.Simó, and J.Solà-Morales, Eds.), World Sci. Publ. Co. Inc., River Edge, NJ, 1993, 1, pp. 3–22.

    MathSciNet  MATH  Google Scholar 

  14. Babin A. B. and Vishik M. I., Attractors of Evolution Equations, Amsterdam etc., North-Holland (1992).

    MATH  Google Scholar 

  15. Feireisl E., “Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems,” J. Dyn. Differ. Equations, 6, 23–35 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  16. Anh C. T., Hung P. Q., Ke T. D., and Phong T. T., “Global attractor for a semilinear parabolic equation involving the Grushin operator,” Electron. J. Differ. Equ., No. 32, 1–11 (2008).

    MathSciNet  MATH  Google Scholar 

  17. Kogoj A. E. and Sonner S., “Attractors met X-elliptic operators,” J. Math. Anal. Appl., 420, 407–434 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  18. Thuy P. T. and Tri N. M., “Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators,” Nonlinear Diff. Equ. Appl., 20, No. 3, 1213–1224 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  19. Chueshov I. D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA Sci. Publ. House, Kharkov, Ukraine (2002).

    MATH  Google Scholar 

  20. Hale J. K., Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence (1988) (Math. Surveys Monogr.; V. 25).

    MATH  Google Scholar 

  21. Raugel G., “Global attractors in partial differential equations,” in: Handbook of Dynamical Systems. Vol. 2, North-Holland, Amsterdam, 2002, pp. 885–892.

    Google Scholar 

  22. Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York (1983) (Appl. Math. Sci.; V. 44).

    Book  MATH  Google Scholar 

  23. Sell G. R. and You Y., Dynamics of Evolutionary Equations, Springer-Verlag, New York (2002).

    Book  MATH  Google Scholar 

  24. Pražák D., “On finite fractal dimension of the global attractor for the wave equation with nonlinear damping,” J. Dyn. Differ. Equations, 14, 763–776 (2002).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hoa Lu University, Ninh Nhat, Vietnam

    D. T. Luyen

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    N. M. Tri

Authors
  1. D. T. Luyen
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  2. N. M. Tri
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Correspondence to N. M. Tri.

Additional information

Original Russian Text Copyright © 2016 Luyen D.T. and Tri N.M.

Ninh Nhat; Hanoi. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 4, pp. 809–829, July–August, 2016; DOI: 10.17377/smzh.2016.57.407. Original article submitted March 20, 2015.

The authors are supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant 101.02–2014.50).

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Luyen, D.T., Tri, N.M. Large-time behavior of solutions to degenerate damped hyperbolic equations. Sib Math J 57, 632–649 (2016). https://doi.org/10.1134/S0037446616040078

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  • DOI: https://doi.org/10.1134/S0037446616040078

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