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Regularity of Invariant Sets in Variable Internal Damped Wave Equations

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Abstract

In this paper we prove that every compact invariant subset \(\mathscr{A}\) associated with the semigroup Sn,k(t)t≥0 generated by wave equations with variable damping, either in the interior or on the boundary of the domain where Ω ⊂ ℝ3 is a smooth bounded domain, in H 10 (Ω) × L2(Ω) is in fact bounded in D(B0) × H 10 (Ω). As an application of our results, we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H1(Ω) × L2(Ω) when the interior variable damping converges to the boundary damping in the sense of distributions.

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References

  1. Babin, A.V., Vishik, M.I. Attractors of evolution equations. Studies in Mathematics and its Applications, North-Holland, 1992

    MATH  Google Scholar 

  2. Ball, J.M. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 10: 31–52 (2004)

    Article  MathSciNet  Google Scholar 

  3. Cazenave, T., Haraux, A. An Introduction to Semilinear Evolution Equations. Clarendon Press, Oxford, 1988

    MATH  Google Scholar 

  4. Conti, M., Pata, V. On the regularity of global attractors. Discrete and Continuous Dynamical Systems, 25: 1209–1217 (2009)

    Article  MathSciNet  Google Scholar 

  5. Chen, G., Fulling, S.A., Narcowich, F.J., Sun, S. Exponential decay of energy of evolution equations with locally distributed damping. SIAM Journal on Applied Mathematics, 51: 266–301 (1991)

    Article  MathSciNet  Google Scholar 

  6. Chueshov, I., Eller, M., Lasiecka, I. On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Communications in Partial Differential Equations, 27: 1901–1951 (2002)

    Article  MathSciNet  Google Scholar 

  7. Cox, S., Zuazua, E. The rate at which energy decays in a damped string. Communication in Partial Differential Equations, 19: 213–243 (1994)

    Article  MathSciNet  Google Scholar 

  8. Cox, S., Zuazua, E. The rate at which energy decays in a string damped at one end. Indiana University Mathematics Journal, 44: 545–573 (1995)

    Article  MathSciNet  Google Scholar 

  9. Eden, A., Milani, A. On the convergence of attractors and exponential attractors for singularly perturbed hyperbolic equations. Turkish J. Math., 19: 102–117 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Feireisel, E., Zuazua, E. Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent. Communications in Partial Differential Equations, 18: 1539–1555 (1993)

    Article  MathSciNet  Google Scholar 

  11. Ghidaglia, J.M., Temam, R. Regularity of the solutions of second order evolution equations and their attractors. Ann. Sc. Norm. Super. Pisa Cl. Sci., 14: 485–511 (1987)

    MathSciNet  MATH  Google Scholar 

  12. Grasselli, M., Pata, V. On the damped semilinear wave equation with critical exponent. In: Dynamical Systems and Differential Equations, Wilmington, NC, 2002, Discrete Contin. Dyn. Syst. (Suppl.), 351–358 (2003)

  13. Hale, J.K. Asymptotic behavior of dissipative systems. Mathematical Survey, American Mathematical Society, 1988

  14. Hale, J.K., Raugel, G. Regularity, determining modes and Galerkin methods. J. Math. Pures Appl., 82: 1075–1136 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Haraux, A. Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugaliae Mathematica, 46: 246–257 (1989)

    MATH  Google Scholar 

  17. Haraux, A. Two remarks on hyperbolic dissipative problems. in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. VII, Paris, 1983–1984, Pitman, Boston, 1985, 161–179.

  18. Joly, R. Convergence of the wave equation damped on the interior to the one damped on the boundary. Journal of Differential Equations, 229: 588–653 (2006)

    Article  MathSciNet  Google Scholar 

  19. Kato, T. Linear evolution equations of hyperbolic type. J. Fac. Sci. Univ. Tokyo Sect., 17: 241–258 (1970)

    MathSciNet  MATH  Google Scholar 

  20. Komornik, V., Zuazua, E. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl., 69: 33–45 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Kostin, I.N. Attractor for a semilinear wave equation with boundary damping. Journal of Mathematical Sciences, 98: 753–764 (2000)

    Article  MathSciNet  Google Scholar 

  22. Lasiecka, I., Tataru, D. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential and Integral Equations, 6: 507–533 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Pazy, A. Semigroups of linear operators and applications to partial differential equa- tions. Applied Mathematical Sciences, Springer-Verlag, 1983

    Book  Google Scholar 

  24. Prizzi, M. Regularity of invariant sets in semilinear damped wave equations. J. Differential Equations, 247: 3315–3337 (2009)

    Article  MathSciNet  Google Scholar 

  25. Rybakowski, K.P. Conley index continuation for singularly perturbed hyperbolic equations. Topol. Methods Nonlinear Anal., 22: 203–244 (2003)

    Article  MathSciNet  Google Scholar 

  26. Rauch, J. Qualitative behavior of dissipative wave equations on bounded domains. Archive of Rational Mechanics and Analysis, 62: 77–85 (1976)

    Article  MathSciNet  Google Scholar 

  27. Tataru, D. Uniform decay rates and attractors for evolution PDEs with boundary dissipation. Journal of Differential Equations, 121: 1–27 (1995)

    Article  MathSciNet  Google Scholar 

  28. Temam, R. Infinite-dimensional systems in mechanics and physics. Springer-Verlag, New York, 1997

    Book  Google Scholar 

  29. Zuazua, E. Exponential decay for the semilinear wave equation with locally distributed damping. Communications in Partial Differential Equations, 15: 205–235 (1990)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors express their sincere thanks to the referee for his/her valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Gao-cheng Yue, Yu-xin Liang & Jia-jia Yang

Authors
  1. Gao-cheng Yue
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  2. Yu-xin Liang
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  3. Jia-jia Yang
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Correspondence to Gao-cheng Yue.

Additional information

The first author is supported by “the Fundamental Research Funds for the Central Universities”, NO.NS2020058 and was supported by the National Natural Science Foundation of China under Grant 11501289.

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Yue, Gc., Liang, Yx. & Yang, Jj. Regularity of Invariant Sets in Variable Internal Damped Wave Equations. Acta Math. Appl. Sin. Engl. Ser. 36, 952–974 (2020). https://doi.org/10.1007/s10255-020-0978-4

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  • DOI: https://doi.org/10.1007/s10255-020-0978-4

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