Semigroup Forum

, Volume 88, Issue 2, pp 333–365 | Cite as

Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions

  • Philip Jameson GraberEmail author
  • Irena Lasiecka


We consider a linear system of PDEs of the form
$$\begin{aligned} & \begin{aligned} u_{tt} - c\Delta u_t - \Delta u &= 0 \quad\text{in } \varOmega\times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\varGamma(c \alpha u_t + u)& = 0 \quad\text{on } \varGamma_1 \times(0,T)\\ u &= 0 \quad\text{on } \varGamma_0 \times(0,T) \end{aligned} \\ &\quad (u(0),u_t(0),u|_{\varGamma_1}(0),u_t|_{\varGamma_1}(0)) \in {\mathcal{H}} \end{aligned}$$
on a bounded domain Ω with boundary Γ=Γ 1Γ 0. We show that the system generates a strongly continuous semigroup T(t) which is analytic for α>0 and of Gevrey class for α=0. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form ∥(d/dt)T(t)∥≲|t|−1 for α>0, ∥(d/dt)T(t)∥≲|t|−2 for α=0. Moreover, when α=0 we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to 1/2 power of the generator.


Gevrey’s semigroups Analytic semigroups Wave equation with dynamic boundary conditions Pseudodifferential operators 


  1. 1.
    Carvalho, A.N., Cholewa, J.W., Dlotko, T.: Strongly damped wave problems: bootstrapping and regularity of solutions. J. Differ. Equ. 244, 2310–2333 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Casarino, V., Engel, K.-J., Nickel, G., Piazzera, S.: Decoupling techinques for wave equations with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 12, 761–772 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Chen, G., Russell, D.L.: A mathematical model for linear elastic systems with structural damping. Tech. rep., DTIC Document (1980) Google Scholar
  4. 4.
    Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136 (1989) Google Scholar
  5. 5.
    Chen, S.P., Triggiani, R.: Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<α<1/2. Proc. Am. Math. Soc. 110, 401–415 (1990) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Engel, K.-J.: Spectral theory and generator property for one-sided coupled operator matrices. Semigroup Forum 58, 267–295 (1999). doi: 10.1007/s002339900020 CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Favini, A., Gal, C.G., Goldstein, J., Romanelli, S., Goldstein, G.R.: The non-autonomous wave equation with general Wentzell boundary conditions. In: Proceedings of the Royal Society of Edinburgh—A-Mathematics, vol. 135, pp. 317–330. Cambridge Univ. Press, Cambridge (2005) Google Scholar
  8. 8.
    Favini, A., Goldstein, G., Goldstein, J., Romanelli, S.: The heat equation with generalized Wentzell boundary conditions. J. Evol. Equ. 2, 1–19 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Favini, A., Goldstein, G.R., Goldstein, J.A., Obrecht, E., Romanelli, S.: Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem. Math. Nachr. 283, 504–521 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Fourrier, N.: Regularity, stability and numerical analysis of a wave equation with strong damping and dynamic boundary conditions. PhD thesis, University of Virginia (2013) Google Scholar
  11. 11.
    Gal, C., Goldstein, G., Goldstein, J.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623–635 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gerbi, S., Said-Houari, B.: Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Anal., Theory Methods Appl. 74, 7137–7150 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Goldstein, G.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11, 457–480 (2006) zbMATHGoogle Scholar
  14. 14.
    Graber, P.J., Said-Houari, B.: Existence and asymptotic behavior of the wave equation with dynamic boundary conditions. Appl. Math. Optim. 42 pp. doi: 10.1007/s00245-012-9165-1
  15. 15.
    Guidetti, D.: Abstract elliptic problems depending on a parameter and parabolic problems with dynamic boundary conditions. Preprint (2013) Google Scholar
  16. 16.
    Haraux, A., Otani, M., et al.: Analyticity and regularity for a class of second order evolution equations. Evol. Equ. Control Theory 2, 101–117 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators vol. 3. Springer, Berlin (2007) Google Scholar
  18. 18.
    Horn, M.A., Littman, W.: Boundary control of a Schrödinger equation with nonconstant principal part. In: Control of partial differential equations and applications, Laredo, 1994. Lecture Notes in Pure and Appl. Math, vol. 174, pp. 101–106 (1994) Google Scholar
  19. 19.
    Huang, F.: On the mathematical model for linear elastic systems with analytic damping. SIAM J. Control Optim. 26, 714–724 (1988) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations. Cambridge University Press, Cambridge (2000) CrossRefGoogle Scholar
  21. 21.
    Lasiecka, I., Triggiani, R., Yao, P.-F.: Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235, 13–57 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lasiecka, I., Triggiani, R., Yao, P.-F.: An observability estimate in L 2(ΩH −1(Ω) for second-order hyperbolic equations with variable coefficients. Control Distribut. Parameter Stoch. Syst. 13, 71 (1999) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Littman, W., Taylor, S.: Smoothing evolution equations and boundary control theory. J. Anal. Math. 59, 117–131 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16. Birkhäuser, Basel (2003) Google Scholar
  25. 25.
    Mugnolo, D.: Second order abstract initial-boundary value problems. PhD thesis, Eberhard-Karls-Universität Tübingen (2004) Google Scholar
  26. 26.
    Mugnolo, D.: Abstract wave equation with acoustic boundary conditions. Math. Nachr. 279, 299–318 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Mugnolo, D.: A variational approach to strongly damped wave equations. In: Functional Analysis and Evolution Equations, pp. 503–514. Springer, Berlin (2008) CrossRefGoogle Scholar
  28. 28.
    Mugnolo, D.: Damped wave equations with dynamic boundary conditions. J. Anal. Appl. 17, 241 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Nagel, R.: Toward a “matrix theory” for unbounded operator matrices. Math. Z. 201, 57–68 (1989) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983) CrossRefzbMATHGoogle Scholar
  31. 31.
    Renardy, M.: On the stability of differentiability of semigroups. Semigroup Forum 51, 343–346 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series. Princeton University Press, Princeton (1981) zbMATHGoogle Scholar
  33. 33.
    Taylor, S.: Gevrey semigroups (Chapter). PhD thesis, Thesis, School of Mathematics, University of Minnesota (1989) Google Scholar
  34. 34.
    Triggiani, R., Yao, P., Lasiecka, I.: Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms. Nonlinear Anal., Theory Methods Appl. 30, 111–122 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Vazquez, J., Vitillaro, E.: Heat equation with dynamical boundary conditions of reactive-diffusive type. J. Differ. Equ. 250, 2143–2161 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Xiao, T.-J., Liang, J.: The Cauchy Problem for Higher Order Abstract Differential Equations. Springer, Berlin (1998) CrossRefzbMATHGoogle Scholar
  37. 37.
    Xiao, T.-J., Liang, J.: Complete second order differential equations in Banach spaces with dynamic boundary conditions. J. Differ. Equ. 200, 105–136 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Xiao, T.-J., Liang, J.: Second order parabolic equations in Banach spaces with dynamic boundary conditions. Trans. Am. Math. Soc. 356, 4787–4810 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Yao, P.-F.: On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37, 1568–1599 (1999) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Unité de Mathématiques AppliquéesCommands (ENSTA ParisTech, INRIA Saclay)Palaiseau CedexFrance
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations