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On a Class of Nonlinear Viscoelastic Kirchhoff Plates: Well-Posedness and General Decay Rates

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Abstract

This paper is concerned with well-posedness and energy decay rates to a class of nonlinear viscoelastic Kirchhoff plates. The problem corresponds to a class of fourth order viscoelastic equations of \(p\)-Laplacian type which is not locally Lipschitz. The only damping effect is given by the memory component. We show that no additional damping is needed to obtain uniqueness in the presence of rotational forces. Then, we show that the general rates of energy decay are similar to ones given by the memory kernel, but generally not with the same speed, mainly when we consider the nonlinear problem with large initial data.

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References

  1. Alabau-Boussouira, F., Cannarsa, P., Sforza, D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254(5), 1342–1372 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrade, D., Silva, J., Ma, T.F.: Exponential stability for a plate equation with p-Laplacian and memory terms. Math. Methods Appl. Sci. 35(4), 417–426 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Araújo, R.O., Ma, T.F., Qin, Y.: Long-time behavior of a quasilinear viscoelastic equation with past history. J. Differ. Equ. 254(10), 4066–4087 (2013)

    Article  MATH  Google Scholar 

  4. Barreto, R., Lapa, E.C., Rivera, J.E.M.: Decay rates for viscoelastic plates with memory. J. Elast. 44(1), 61–87 (1996)

    Article  MATH  Google Scholar 

  5. Cavalcanti, M.M., Domingos, V.N., Ferreira, J.: Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24(14), 1043–1053 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cavalcanti, M.M., Oquendo, H.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cavalcanti, M.M., Cavalcanti, V.N.D., Ma, T.F.: Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler–Bernoulli equation with a nonlocal dissipation in general domains. Differ. Integr. Equ. 17, 495–510 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Cavalcanti, M.M., Cavalcanti, A.D.D., Lasiecka, I., Wang, X.: Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal. RWA 22, 289–306 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fabrizio, M., Giorgi, C., Pata, V.: A new approach to equations with memory. Arch. Ration. Mech. Anal. 198, 189–232 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giorgi, C., Rivera, J.E.M., Pata, V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260(1), 83–99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Han, X., Wang, M.: General decay of energy for a viscoelastic equation with nonlinear damping. Math. Methods Appl. Sci. 32(3), 346–358 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Han, X., Wang, M.: General decay of energy for a viscoelastic equation with nonlinear damping. J. Franklin Inst. 347(5), 806–817 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lagnese, J.E.: Asymptotic Energy Estimates for Kirchhoff Plates Subject to Weak Viscoelastic Damping. Control and Estimation of Distributed Parameter Systems (Vorau, 1988), pp. 211–236, Internat. Ser. Numer. Math., 91. Birkhäuser, Basel (1989)

  15. Lagnese, J.E.: Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, vol. 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1989)

    Book  Google Scholar 

  16. Lagnese, J., Lions, J.-L.: Modelling, Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées, vol. 6. Masson, Paris (1988)

    Google Scholar 

  17. Lasiecka, I., Messaoudi, S.A., Mustafa, M.: Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 54(3), 031504 (2013)

    Article  MathSciNet  Google Scholar 

  18. Liu, W.: General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys. 50(11), 113506 (2009)

    Article  MathSciNet  Google Scholar 

  19. Liu, W.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73, 1890–1904 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu, W., Sun, Y.: General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions. Z. Angew. Math. Phys. 65(1), 125–134 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Messaoudi, S.A.: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341(2), 1457–1467 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Messaoudi, S.A.: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal. 69(8), 2589–2598 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Messaoudi, S.A., Tatar, N.-E.: Exponential decay for a quasilinear viscoelastic equation. Math. Nachr. 282, 1443–1450 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Messaoudi, S.A., Mustafa, M.I.: General stability result for viscoelastic wave equations. J. Math. Phys. 53(5), 053702 (2012)

    Article  MathSciNet  Google Scholar 

  25. Park, J.Y., Park, S.H.: General decay for quasilinear viscoelastic equations with nonlinear weak damping. J. Math. Phys. 50, 083505 (2009)

    Article  MathSciNet  Google Scholar 

  26. Pata, V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 64(3), 499–513 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pata, V.: Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77, 333–360 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rivera, J.E.M., Salvatierra, A.P.: Asymptotic behaviour of the energy in partially viscoelastic materials. Quart. Appl. Math. 59(3), 557–578 (2001)

    MATH  MathSciNet  Google Scholar 

  29. Rivera, J.E.M., Naso, M.G.: Asymptotic stability of semigroups associated with linear weak dissipative systems with memory. J. Math. Anal. Appl. 326(1), 691–707 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Silva, M.A.J., Ma, T.F.: On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type. IMA J. Appl. Math. 78(6), 1130–1146 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Silva, M.A.J., Ma, T.F.: Long-time dynamics for a class of Kirchhoff models with memory. J. Math. Phys. 54(2), 021505 (2013)

    Article  MathSciNet  Google Scholar 

  32. Tatar, N.E.: Arbitrary decays in linear viscoelasticity. J. Math. Phys. 52(1), 013502 (2011)

    Article  MathSciNet  Google Scholar 

  33. Wu, S.T.: Arbitrary decays for a viscoelastic equation. Bound. Value Probl. 2011(28), 1–14 (2011)

    Google Scholar 

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Acknowledgments

The authors have been supported by the Brazilian Agency CNPq within the Project ”Ciências sem Fronteiras”, Grant #402689/2012-7. The first author was also partially supported by the CNPq Grant #441414/2014-1.

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Authors and Affiliations

  1. Department of Mathematics, State University of Londrina, Londrina, PR, 86057-970, Brazil

    M. A. Jorge Silva

  2. National Laboratory of Scientific Computation, Petrópolis, RJ, 25651-070, Brazil

    J. E. Muñoz Rivera

  3. Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, 21941-909, Brazil

    J. E. Muñoz Rivera

  4. Department of Mathematics and Statistics, University of Konstanz, 78457, Konstanz, Germany

    R. Racke

Authors
  1. M. A. Jorge Silva
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  2. J. E. Muñoz Rivera
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  3. R. Racke
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Correspondence to M. A. Jorge Silva.

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Silva, M.A.J., Rivera, J.E.M. & Racke, R. On a Class of Nonlinear Viscoelastic Kirchhoff Plates: Well-Posedness and General Decay Rates. Appl Math Optim 73, 165–194 (2016). https://doi.org/10.1007/s00245-015-9298-0

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