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Stochastic nonlinear beam equations
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  • Published: 11 November 2004

Stochastic nonlinear beam equations

  • Zdzisław Brzeźniak1,
  • Bohdan Maslowski2 &
  • Jan Seidler2 

Probability Theory and Related Fields volume 132, pages 119–149 (2005)Cite this article

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Abstract.

An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution.

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References

  1. Ball, J.M.: Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 42, 61–90 (1973)

    Article  MATH  Google Scholar 

  2. Brzeźniak, Z.: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61, 245–295 (1997)

    MathSciNet  Google Scholar 

  3. Brzeźniak, Z., Gatarek, D.: Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces. Stochastic Process. Appl. 84, 187–225 (1999)

    Article  Google Scholar 

  4. Carroll, A.: The stochastic nonlinear heat equation. Ph. D. Thesis. The University of Hull, 1999

  5. Caraballo, T., Liu, K., Mao, X.: On stabilization of partial differential equations by noise. Nagoya Math. J, 161, 155–170 (2001)

    Google Scholar 

  6. Cazenave, T., Haraux, A.: An introduction to semilinear evolution equations. Clarendon Press, Oxford, 1998

  7. Ceron, S.S., Lopes, O.: α-contractions and attractors for dissipative semilinear hyperbolic equations and systems. Ann. Math. Pura Appl. 160 (4), 193–206 (1991)

    MATH  Google Scholar 

  8. Chow, P.L., Menaldi, J.L.: Stochastic PDE for nonlinear vibration of elastic panels. Differential Integral Equations 12, 419–434 (1999)

    MATH  Google Scholar 

  9. Chueshov, I.D.: Introduction to the theory of infinite-dimensional dissipative systems, Acta Kharkiv, 2002

  10. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge Univ. Press, Cambridge, 1992

  11. Dickey, R.W.: Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29, 443–454 (1970)

    Article  MATH  Google Scholar 

  12. Dudley, R.M.: Real analysis and probability. Wadsworth & Brooks/Cole, Pacific Grove, 1989

  13. Eden, A., Milani, A.J.: Exponential attractors for extensible beam equations. Nonlinearity 6, 457–479 (1993)

    Article  MATH  Google Scholar 

  14. Eisley, J.G.: Nonlinear vibration of beams and rectangular plates. Z. Angew. Math. Phys. 15, 167–175 (1964)

    MATH  Google Scholar 

  15. Feireisl, E.: Finite-dimensional behaviour of a non-autonomous partial differential equation: forced oscillations of an extensible beam. J. Differential Equations 101, 302–312 (1993)

    Article  MATH  Google Scholar 

  16. Fitzgibbon, W.E.: Global existence and boundedness of solutions to the extensible beam equation. SIAM J. Math. Anal. 13, 739–745 (1982)

    MATH  Google Scholar 

  17. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin, 1983

  18. Holmes, P., Marsden, J.: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Rational Mech. Anal. 76, 135–165 (1981)

    Article  MATH  Google Scholar 

  19. Khas’minskii, R.Z.: Stability of systems of differential equations under random perturbations of their parameters. Nauka, Moskva, 1969 (Russian); Stochastic stability of differential equations, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980

  20. Kouémou Patcheu, S.: On a global solution and asymptotic behaviour for the generalized damped extensible beam equation. J. Differential Equations 135, 299–314 (1997)

    Article  MATH  Google Scholar 

  21. Leha, G., Maslowski, B., Ritter, G.: Stability of solutions to semilinear stochastic evolution equations. Stochastic Anal. Appl. 17, 1009–1051 (1999)

    MATH  Google Scholar 

  22. Liu, K., Truman, A.: Lyapunov function approaches and asymptotic stability of stochastic evolution equations in Hilbert spaces –- a survey of recent developments. Stochastic partial differential equations and applications (Trento, 2000), Dekker, New York, 2002, pp. 337–371

  23. Maslowski, B., Seidler, J., Vrkoč, I.: Integral continuity and stability for stochastic hyperbolic equations. Differential Integral Equations 6, 355–382 (1993)

    MATH  Google Scholar 

  24. Medeiros, L.A.: On a new class of nonlinear wave equations. J. Math. Anal. Appl. 69, 252–262 (1979)

    Article  MATH  Google Scholar 

  25. Ondreját, M.: Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. 426, 1–63 (2004)

    Google Scholar 

  26. Ondreját, M.: Brownian representation of cylindrical local martingales, martingale problem and strong Markov property of weak solutions of SPDEs in Banach spaces. Czechoslovak Math. J. (to appear)

  27. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, 1983

  28. Pritchard, A.J., Zabczyk J.: Stability and stabilizability of infinite dimensional systems. SIAM Review 23, 25–52 (1981)

    MATH  Google Scholar 

  29. Reiss, E.L., Matkowsky, B.J.: Nonlinear dynamic buckling of a compressed elastic column. Q. Appl. Math. 29, 245–260 (1971)

    MATH  Google Scholar 

  30. Seidler, J.: Da Prato-Zabczyk’s maximal inequality revisited I. Math. Bohem. 118, 67–106 (1993)

    MATH  Google Scholar 

  31. Triebel, H.: Interpolation theory, function spaces, differential operators. Deutscher Verlag der Wissenschaften, Berlin, 1978

  32. Unai, A.: Abstract nonlinear beam equations. SUT J. Math. 29, 323–336 (1993)

    MATH  Google Scholar 

  33. Vasconcellos, C.F., Teixeira, L.M.: Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping. Ann. Fac. Sci. Toulouse Math. 8 (6), 173–193 (1999)

    MATH  Google Scholar 

  34. Woinowsky-Krieger, S.: The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Hull, Hull, HU6 7RX, United Kingdom

    Zdzisław Brzeźniak

  2. Mathematical Institute, Academy od Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Bohdan Maslowski & Jan Seidler

Authors
  1. Zdzisław Brzeźniak
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  2. Bohdan Maslowski
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  3. Jan Seidler
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Corresponding author

Correspondence to Jan Seidler.

Additional information

This research was supported in part by the GA ČR Grants no. 201/98/1454, 201/01/1197 and by a Royal Society grant

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Cite this article

Brzeźniak, Z., Maslowski, B. & Seidler, J. Stochastic nonlinear beam equations. Probab. Theory Relat. Fields 132, 119–149 (2005). https://doi.org/10.1007/s00440-004-0392-5

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  • Received: 09 June 2004

  • Revised: 02 September 2004

  • Published: 11 November 2004

  • Issue Date: May 2005

  • DOI: https://doi.org/10.1007/s00440-004-0392-5

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Key words

  • Stochastic extensible beam equation
  • Lyapunov functions
  • Nonexplosion
  • Exponential stability
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