Advertisement

Applied Mathematics & Optimization

, Volume 66, Issue 1, pp 81–122 | Cite as

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

  • Philip Jameson Graber
  • Belkacem Said-HouariEmail author
Article

Abstract

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.

Keywords

Wave equation Dynamic boundary condition Source Damping Blow up Finite time Exponential growth 

Notes

Acknowledgements

The first author wishes to thank the Virginia Space Grant Consortium and the Jefferson Scholars Foundation for their support. The second author wants to thank KAUST for its support. Both authors are very grateful to Prof. Irena Lasiecka for many fruitful discussions.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) zbMATHGoogle Scholar
  2. 2.
    Ahmed, N.U., Skowronski, J.M.: Stability and control of nonlinear flexible systems. Dyn. Syst. Appl. 2(2), 149–162 (1993) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andrews, K.T., Kuttler, K.L., Shillor, M.: Second order evolution equations with dynamic boundary conditions. J. Math. Anal. Appl. 197(3), 781–795 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Groningen (1976) CrossRefzbMATHGoogle Scholar
  5. 5.
    Beale, J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25(9), 895–917 (1976) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bociu, L., Lasiecka, I.: Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping. Discrete Contin. Dyn. Syst. 22(4), 835–860 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bociu, L., Lasiecka, I.: Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping. J. Differ. Equ. 249, 654–683 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972) MathSciNetGoogle Scholar
  9. 9.
    Budak, B.M., Samarskii, A.A., Tikhonov, A.N.: A Collection of Problems on Mathematical Physics. Macmillan Co., New York (1964). Translated by A.R.M. Robson Google Scholar
  10. 10.
    Caroll, R.W., Showalter, R.E.: Singular and Degenerate Cauchy Problems. Academic Press, New York (1976) Google Scholar
  11. 11.
    Cavalcanti, M.M., Cavalcanti, V.D., Lasiecka, I.: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differ. Equ. 236, 407–459 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136(1), 15–55 (1989) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chueshov, I., Eller, M., Lasiecka, I.: On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun. Partial Differ. Equ. 27(9–10), 1901–1951 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Conrad, F., Morgul, O.: Stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 36(6), 1962–1986 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Favini, A., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: The heat equation with generalized Wentzell boundary condition. J. Evol. Equ. 2, 1–19 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gal, C.G., Goldstein, G.R., Goldstein, J.A.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623–635 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gazzola, F., Squassina, M.: Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. Henri Poincaré 23, 185–207 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grobbelaar-Van Dalsen, M.: On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions. Appl. Anal. 53(1–2), 41–54 (1994) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Grobbelaar-Van Dalsen, M.: On the initial-boundary-value problem for the extensible beam with attached load. Math. Methods Appl. Sci. 19(12), 943–957 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Georgiev, V., Todorova, G.: Existence of a solution of the wave equation with nonlinear damping and source term. J. Differ. Equ. 109, 295–308 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gerbi, S., Said-Houari, B.: Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Adv. Differ. Equ. 13(11–12), 1051–1074 (2008) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gerbi, S., Said-Houari, B.: Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Anal. 74, 7137–7150 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ruiz Goldstein, G.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11(4), 457–480 (2006) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Graber, P.J.: Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system. Nonlinear Anal. 73, 3058–3068 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Graber, P.J.: Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions. Nonlinear Anal. 74, 3137–3148 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507–533 (1993) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Springer, New York (1972) Google Scholar
  28. 28.
    Littman, W., Markus, L.: Stabilization of a hybrid system of elasticity by feedback boundary damping. Ann. Mat. Pura Appl., IV. Ser. 152, 281–330 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Messaoudi, S., Said-Houari, B.: Global non-existence of solutions of a class of wave equations with non-linear damping and source terms. Math. Methods Appl. Sci. 27, 1687–1696 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Meurer, T., Kugi, A.: Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator. Int. J. Robust Nonlinear Control 21, 542–562 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mifdal, A.: Stabilisation uniforme d’un système hybride. C. R. Acad. Sci., Ser. 1 Math. 324(1), 37–42 (1997) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Mugnolo, D.: Damped wave equations with dynamic boundary conditions. J. Appl. Anal. 17(2), 241–275 (2011) CrossRefGoogle Scholar
  33. 33.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) CrossRefzbMATHGoogle Scholar
  34. 34.
    Pellicer, M.: Large time dynamics of a nonlinear spring-mass-damper model. Nonlinear Anal. 69(1), 3110–3127 (2008) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Pellicer, M., Solà-Morales, J.: Analysis of a viscoelastic spring-mass model. J. Math. Anal. Appl. 294(2), 687–698 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pellicer, M., Solà-Morales, J.: Spectral analysis and limit behaviours in a spring-mass system. Commun. Pure Appl. Anal. 7(3), 563–577 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rao, B.: Decay estimates of solutions for a hybrid system of flexible structures. Eur. J. Appl. Math. 4(3), 303–319 (1993) CrossRefzbMATHGoogle Scholar
  38. 38.
    Rao, B.: Stabilisation du modèle SCOLE par un contrôle a priori borné. C. R. Acad. Sci., Ser. 1 Math. 316(10), 1061–1066 (1993) zbMATHGoogle Scholar
  39. 39.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. AMS, Providence (1997) zbMATHGoogle Scholar
  40. 40.
    Vitillaro, E.: Global existence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155–182 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    You, Y.C., Lee, E.B.: Controllability and stabilization of two-dimensional elastic vibration with dynamical boundary control. In: Control of Partial Differential Equations. Lecture Notes in Control and Information Sciences, vol. 114, pp. 297–308 (1989) CrossRefGoogle Scholar
  42. 42.
    Yu, S.: On the strongly damped wave equation with nonlinear damping and source terms. Electron. J. Qual. Theory Differ. Equ. 39, 1–18 (2009) Google Scholar
  43. 43.
    Zhang, H., Hu, Q.: Energy decay for a nonlinear viscoelastic rod equations with dynamic boundary conditions. Math. Methods Appl. Sci. 30(3), 249–256 (2007) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Division of Mathematical and Computer Sciences and EngineeringKing Abdullah University of Science and Technology (KAUST)ThuwalKingdom of Saudi Arabia

Personalised recommendations