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Plates with Boundary Damping

  • Igor Chueshov
  • Irena Lasiecka
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we consider models where the main source of dissipation is generated by boundary damping. Typical physical examples include boundary friction applied to an edge of the plate. This type of problem—with geometrically constrained support of the damping—is mathematically more demanding. Indeed, the dissipation is active on a much more restrictive support, hence potentially less effective. However, recent developments in the area of boundary stabilization of plates provide new tools in order to carry out the appropriate estimates. These are based on a multipliers method and microlocal analysis estimates. We first consider models with the rotational inertia included. These problems are more tractable and simpler, due to the fact that the nonlinear term acts upon the system as a compact perturbation. Proving asymptotic smoothness in that case is an easier task. When the rotational inertia are absent in the model, the analysis is more involved and delicate, because it involves the need for compensating the lack of compactness.

Keywords

Strong Solution Global Attractor Rotational Inertia Free Boundary Condition Energy Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  1. 63.
    I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Commun. Partial Diff. Eqs., 29 (2004), 1847–1976.MATHCrossRefMathSciNetGoogle Scholar
  2. 68.
    I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Diff. Eqs., 198 (2004), 196–221.MATHCrossRefMathSciNetGoogle Scholar
  3. 69.
    I. Chueshov and I. Lasiecka, Attractors for second order evolution equations, J. Dynam. Diff. Eqs., 16 (2004), 469–512.MATHCrossRefMathSciNetGoogle Scholar
  4. 73.
    I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping, J. Diff. Eqs., 233 (2007), 42–86.MATHCrossRefMathSciNetGoogle Scholar
  5. 75.
    I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008.Google Scholar
  6. 78.
    I. Chueshov, I. Lasiecka, and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discr. Cont. Dyn. Sys., 20 (2008), 459–509.MATHMathSciNetGoogle Scholar
  7. 79.
    I. Chueshov, I. Lasiecka, and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Diff. Eqs., 21 (2009), 269–314.MATHCrossRefMathSciNetGoogle Scholar
  8. 85.
    P. Ciarlet and P. Rabier, Les Equations de von Karman, Springer, Berlin, 1980.MATHGoogle Scholar
  9. 151.
    G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behaviour, J. Math. Anal. Appl., 229 (1999), 452–479.MATHCrossRefMathSciNetGoogle Scholar
  10. 160.
    A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear dissipation, J. Math. Anal. Appl., 318 (2006), 92–101.MATHCrossRefMathSciNetGoogle Scholar
  11. 161.
    A. K. Khanmamedov, Finite dimensionality of the global attractors to von Karman equations with nonlinear interior dissipation, Nonlin. Anal., 66 (2007), 204–213.MATHCrossRefMathSciNetGoogle Scholar
  12. 173.
    J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, 1989.MATHGoogle Scholar
  13. 176.
    J. Lagnese, Modeling and stabilization of nonlinear plates, Int. Ser. Num. Math., 100 (1991), 247–264.MathSciNetGoogle Scholar
  14. 185.
    I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, J. Math. Pure Appl., 78 (1999), 203–232.MATHMathSciNetGoogle Scholar
  15. 195.
    I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation. Diff. Integral Eqs., 6 (1993), 507–533.MATHMathSciNetGoogle Scholar
  16. 204.
    I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli equations, Appl. Math. Optim., 28 (1993), 277–306.MATHCrossRefMathSciNetGoogle Scholar
  17. 226.
    J.Málek and D. Pražak, Large time behavior via the method of l-tra-jectories, J. Diff. Eqs., 181 (2002), 243–279.MATHCrossRefGoogle Scholar
  18. 244.
    D. Pražak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dyn. Diff. Eqs., 14 (2002), 764–776.Google Scholar
  19. 269.
    D. Tataru, A priori estimates of Carleman’s type in domains with boundary, J. Math. Pures Appl., 73 (1994), 355–387.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media 2010

Authors and Affiliations

  1. 1.Department of Mechanics & MathematicsKharkov National UniversityKharkovUkraine
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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