Skip to main content
SpringerLink
Log in

Polynomial Decay Rate for Dissipative Wave Equations with Mixed Boundary Conditions

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

By using Fourier-Bessel analysis combined with Ingham’s inequality and the multiplier techniques, a polynomial decay rate for the energy of the wave equation, in a two-dimensional bounded domain with Wentzell-Dirichlet boundary conditions, is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    MATH  Google Scholar 

  2. Akil, M., Wehbe, A.: Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric condition. Math. Control Relat. Fields 9(1), 97–116 (2019). https://doi.org/10.3934/mcrf.2019005

    Article  MathSciNet  MATH  Google Scholar 

  3. Alabau, F., Cannarsa, P., Komornik, V.: Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2, 127–150 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Ammari, K., Gerbi, S.: Interior feedback stabilisation of wave equations with dynamic boundary delay. Z. Anal. Anwend. 36(03), 297–327 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Ammari, K., Liu, Z., Tucsnak, M.: Decay rates for a beam with pointwise force and moment feedback. Math. Control Signals Syst. 15, 229–255 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Bassam, M., Mercier, D., Nicaise, S., Wehbe, A.: Polynomial stability of the Timoshenko system by one boundary damping. J. Math. Anal. Appl. 425(2), 1177–1203 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Batkai, A., Engel, K.J.: Abstract wave equations with generalized Wentzell boundary conditions. J. Differ. Equ. 207, 1–20 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Batkai, A., Engel, K.J.: Polynomial stability of operator semigroups. Math. Nachr. 279(13–14), 1425–1440 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Bendali, A., Lemrabet, K.: The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation. SIAM J. Appl. Math. 56(6), 1664–1693 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Beniani, A., Benaissa, A., Zennir, K.: Polynomial decay of solutions to the Cauchy problem for a Petrovsky-Petrovsky system in \(\mathbb{R} ^{n}\). Acta Appl. Math. 146, 67–79 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Bonnaillie-Noel, V., Dambrine, M., Hérau, F., Vial, G.: On generalized Ventcel’s type boundary conditions for Laplace operator in a bounded domain. SIAM J. Math. Anal. 42(2), 931–945 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347(2), 455–478 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Boukhatem, Y., Benabderrahmane, B.: Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions. Acta Math. Sin. Engl. Ser. 32(2), 153–174 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1st edn., vol. 5. North Holland, Amsterdam (1973)

    MATH  Google Scholar 

  15. Buffe, R.: Stabilization of the wave equation with Ventcel boundary conditions. J. Math. Pures Appl. 108, 207–259 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Cavalcanti, M.M., Lasiecka, I., Toundykov, D.: Geometrically constrained stabilization of wave equations with Wentzell boundary conditions. Appl. Anal. 91(8), 1427–1452 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Coclite, G.M., Favini, A., Goldstein, G.R., Goldstein, J.A., Obrecht, E., Romanelli, S.: The role of Wentzell boundary conditions in linear and nonlinear analysis. In: Advances in Nonlinear Analysis: Theory Methods and Applications. Math. Probl. Eng. Aerosp. Sci., vol. 3, pp. 277–289. Camb. Sci. Publ., Cambridge (2009)

    Google Scholar 

  18. Diaz, J.I., Tello, L.: On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete Contin. Dyn. Syst., Ser. S 2(2), 253–262 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Duyckaerts, T.: Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface. Asymptot. Anal. 51, 17–45 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Feller, W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55(2), 468–519 (1952)

    MathSciNet  MATH  Google Scholar 

  21. Feller, W.: Generalized second order differential operators and their lateral conditions. Ill. J. Math. 1, 459–504 (1957)

    MathSciNet  MATH  Google Scholar 

  22. Guesmia, A.: Asymptotic behavior for coupled abstract evolution equations with one infinite memory. Appl. Anal. 94(1), 184–217 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Heminna, A.: Stabilisation de problemes de Ventcel. C. R. Acad. Sci. Paris, Sér. I 328, 1171–1174 (1999)

    MathSciNet  MATH  Google Scholar 

  24. Heminna, A.: Stabilisation frontière de problèmes de Ventcel. ESAIM Control Optim. Calc. Var. 5, 591–622 (2000) (electronic)

    MathSciNet  MATH  Google Scholar 

  25. Ingham, A.E.: Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41, 367–369 (1936)

    MathSciNet  MATH  Google Scholar 

  26. Ismailov, M.I.: Inverse source problem for heat equation with nonlocal Wentzell boundary condition. Results Math. 73(68), (2018). https://doi.org/10.1007/s00025-018-0829-2

  27. Komornik, V.: Exact Controllability and Stabilization, the Multiplier Method. RMA, vol. 36. Masson, Paris (1994)

    MATH  Google Scholar 

  28. Li, C., Liang, J., Xiao, T.-J.: Dynamical behaviors of solutions to nonlinear wave equations with vanishing local damping and Wentzell boundary conditions. Z. Angew. Math. Phys. 69, 102 (2018). https://doi.org/10.1007/s00033-018-0996-8

    Article  MathSciNet  MATH  Google Scholar 

  29. Littman, W., Liu, B.: On the spectral properties and stabilization of acoustic flow. SIAM J. Appl. Math. 59(1), 17–34 (1999) (electronic)

    MathSciNet  MATH  Google Scholar 

  30. Liu, Z., Rao, B.: Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56(4), 630–644 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Liu, C., Wu, H.: An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal. (2019). https://doi.org/10.1007/s00205-019-01356-x

    Article  MathSciNet  MATH  Google Scholar 

  32. Muñoz Rivera, J., Qin, Y.: Polynomial decay for the energy with an acoustic boundary condition. Appl. Math. Lett. 16(2), 249–256 (2003)

    MathSciNet  MATH  Google Scholar 

  33. Nicaise, S.: Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications. Rend. Mat. Appl. (7) 23, 83–116 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Nicaise, S., Laoubi, K.: Polynomial stabilization of the wave equation with Ventcel’s boundary conditions. Math. Nachr. 283(10), 1428–1438 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Nicaise, S., Li, H., Mazzucato, A.: Regularity and a priori error analysis of a Ventcel problem in polyhedral domains (2016). arXiv:1606.04423v1 [math.AP]

  36. Rao, B.: Stabilization of elastic plates with dynamical boundary control. SIAM J. Control Optim. 36(1), 148–163 (1998) (electronic)

    MathSciNet  MATH  Google Scholar 

  37. Ventcel, A.D.: On boundary conditions for multi-dimensional diffusion processes. Theory Probab. Appl. 4, 164–177 (1959)

    MathSciNet  MATH  Google Scholar 

  38. Watson, G.N.: Atreatise on the Theory of Bessel Functions p. 1922. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  39. Zhang, X., Zuazua, E.: Decay of solutions of the system of thermoelasticity of type III. Commun. Contemp. Math. 5, 25–83 (2003)

    MathSciNet  MATH  Google Scholar 

  40. Zhang, X., Zuazua, E.: Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system. J. Differ. Equ. 204, 380–438 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Zhang, X., Zuazua, E.: Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal. 184, 49–120 (2007)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dynamic of Engines and Vibroacoustic Laboratory, Faculty of Engineer’s Sciences, Boumerdes University, Boumerdes, Algeria

    K. Laoubi & D. Seba

Authors
  1. K. Laoubi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Seba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Seba.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Laoubi, K., Seba, D. Polynomial Decay Rate for Dissipative Wave Equations with Mixed Boundary Conditions. Acta Appl Math 169, 629–646 (2020). https://doi.org/10.1007/s10440-020-00315-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-020-00315-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation