# American Institute of Mathematical Sciences

December  2013, 2(4): 631-667. doi: 10.3934/eect.2013.2.631

## Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions

 1 Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States 2 Department of Mathematics, University of Memphis, Memphis, TN 38152-3370, IBS, Polish Academy of Sciences, Warsaw, Poland

Received  April 2013 Revised  September 2013 Published  November 2013

We present an analysis of regularity and stability of solutions corresponding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system. We shall investigate these properties in the context of wave equation with the damping affecting either the interior dynamics or the boundary dynamics or both.
This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well-posedness of finite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed.
The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of various types of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain.
Citation: Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations and Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
##### References:
 [1] B. Andràs and K. J Engel, Abstract wave equations with generalized Wentzell boundary conditions, Journal of Differential Equations, 207 (2004), 1-20. doi: 10.1016/j.jde.2003.12.005. [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3. [3] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292. doi: 10.1007/PL00005977. [4] G. Avalos and D. Toundykov, Boundary stabilization of structural acoustic interactions with interface on a Reissner-Mindlin plate, Nonlinear Anal. Real World Appl., 12 (2011), 2985-3013. doi: 10.1016/j.nonrwa.2011.05.001. [5] J. T. Beale and S. I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc., 80 (1974), 1276-1278. doi: 10.1090/S0002-9904-1974-13714-6. [6] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917. doi: 10.1512/iumj.1976.25.25071. [7] J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 26 (1977), 199-222. doi: 10.1512/iumj.1977.26.26015. [8] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994. [9] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [10] S. Čanić and A. Mikelić, Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries, SIAM J. Appl. Dyn. Syst., 2 (2003), 431-463. doi: 10.1137/S1111111102411286. [11] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [12] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, Journal of Evolution Equations, 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. [13] A. Favini, C. Gal, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The non-autonomous wave equation with general Wentzell boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 135 (2005), 317-329. doi: 10.1017/S0308210500003905. [14] A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086. [15] B. Friedman, Principles and Techniques of Applied Mathematics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1956. [16] C. Gal, G. R. Goldstein and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, Journal of Evolution Equations, 3 (2003), 623-635. doi: 10.1007/s00028-003-0113-z. [17] S. Gerbi and B. Said-Houari, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Anal., 74 (2011), 7137-7150. doi: 10.1016/j.na.2011.07.026. [18] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), 1051-1074. [19] P. J. Graber and B. Said-Houari, Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122. doi: 10.1007/s00245-012-9165-1. [20] P. J. Graber, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, Journal of Evolution Equations, 12 (2012), 141-164. doi: 10.1007/s00028-011-0127-x. [21] A. Haraux and M. Otani, Analyticity and regularity for a class of second order evolution equations, Evolution Equations and Control Theory, 2 (2013), 101-117. doi: 10.3934/eect.2013.2.101. [22] M. A. Horn and W. Littman, Local smoothing properties of a Schrödinger equation with nonconstant principal part, In: Modelling and optimization of distributed parameter systems, (1996), New York, 104-110. [23] M. A. Horn and W. Littman, Boundary control of a Schrödinger equation with nonconstant principal part, In: Control of partial differential equations and applications, Lecture Notes in Pure and Appl. Math., 174 (1996), New York, 101-106. [24] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [25] W. Littman, The wave operator and $L_p$ norms, J. Math. Mech., 12 (1963), 55-68. [26] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Archive for Rational Mechanics and Analysis, 103 (1988), 193-236. doi: 10.1007/BF00251758. [27] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Annali di Matematica Pura ed Applicata, 152 (1988), 281-330. doi: 10.1007/BF01766154. [28] W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory, J. Anal. Math., 59 (1992), 117-131. doi: 10.1007/BF02790221. [29] W. Littman and S. Taylor, Local smoothing and energy decay for a semi-infinite beam pinned at several points, and applications to boundary control, In: Differential equations, dynamical systems, and control science, Lecture Notes in Pure and Appl. Math., 152 (1994), New York, 683-704. [30] W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), 17-34. doi: 10.1137/S0036139996314106. [31] W. Littman and S. Taylor, The heat and Schrödinger equations: Boundary control with one shot, In: Control Methods in PDE-dynamical Systems, Contemp. Math., 426 (2007), 293-305. doi: 10.1090/conm/426/08194. [32] G. Lumer and R. S. Phillips, On the spectral properties and stabilization of acoustic flow, Pacific J. Math., 11 (1961), 679-698. doi: 10.2140/pjm.1961.11.679. [33] T. Meurer and A. Kugi, Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator, International Journal of Robust and Nonlinear Control, 21 (2011), 542-562. doi: 10.1002/rnc.1611. [34] P. M. Morse and K. U. Ingard, Theoretical Acoustics, Princeton University Press, 1987. [35] D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr., 279 (2006), 299-318. doi: 10.1002/mana.200310362. [36] D. Mugnolo, Damped wave equations with dynamic boundary conditions, J. Appl. Anal., 17 (2011), 241-275. doi: 10.1515/JAA.2011.015. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [38] M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum, 51 (1995), 343-346. doi: 10.1007/BF02573642. [39] S. Taylor, Gevrey's Semigroups, Ph.D. Thesis, University of Minnesota, School of Mathematics 1989. [40] S. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobolev spaces, J. Math. Anal. Appl., 194 (1995), 14-38. doi: 10.1006/jmaa.1995.1284. [41] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second edition. Springer Series in Computational Mathematics, 25. Springer-Verlag, Berlin, 2006. [42] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, In: Operator methods for optimal control problems, Lecture Notes in Pure and Appl. Math., 108 (1987), 283-310. [43] R. P. Vito and S. A. Dixon, Blood Vessel Constitutive Models, Annual Review of Biomedical Engineering, 5 (2003), 413-439. [44] T. J. Xiao and J. Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions, Mathematische Annalen, 327 (2003), 351-363. doi: 10.1007/s00208-003-0457-2. [45] T. J. Xiao and J. Liang, Complete second order differential equations in Banach spaces with dynamic boundary condition, J. Differential Equations, 200 (2004), 105-136. doi: 10.1016/j.jde.2004.01.011. [46] T. J. Xiao and J. Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc., 356 (2004), 4787-4809. doi: 10.1090/S0002-9947-04-03704-3.

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##### References:
 [1] B. Andràs and K. J Engel, Abstract wave equations with generalized Wentzell boundary conditions, Journal of Differential Equations, 207 (2004), 1-20. doi: 10.1016/j.jde.2003.12.005. [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3. [3] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292. doi: 10.1007/PL00005977. [4] G. Avalos and D. Toundykov, Boundary stabilization of structural acoustic interactions with interface on a Reissner-Mindlin plate, Nonlinear Anal. Real World Appl., 12 (2011), 2985-3013. doi: 10.1016/j.nonrwa.2011.05.001. [5] J. T. Beale and S. I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc., 80 (1974), 1276-1278. doi: 10.1090/S0002-9904-1974-13714-6. [6] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917. doi: 10.1512/iumj.1976.25.25071. [7] J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 26 (1977), 199-222. doi: 10.1512/iumj.1977.26.26015. [8] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994. [9] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [10] S. Čanić and A. Mikelić, Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries, SIAM J. Appl. Dyn. Syst., 2 (2003), 431-463. doi: 10.1137/S1111111102411286. [11] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [12] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, Journal of Evolution Equations, 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. [13] A. Favini, C. Gal, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The non-autonomous wave equation with general Wentzell boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 135 (2005), 317-329. doi: 10.1017/S0308210500003905. [14] A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086. [15] B. Friedman, Principles and Techniques of Applied Mathematics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1956. [16] C. Gal, G. R. Goldstein and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, Journal of Evolution Equations, 3 (2003), 623-635. doi: 10.1007/s00028-003-0113-z. [17] S. Gerbi and B. Said-Houari, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Anal., 74 (2011), 7137-7150. doi: 10.1016/j.na.2011.07.026. [18] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), 1051-1074. [19] P. J. Graber and B. Said-Houari, Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122. doi: 10.1007/s00245-012-9165-1. [20] P. J. Graber, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, Journal of Evolution Equations, 12 (2012), 141-164. doi: 10.1007/s00028-011-0127-x. [21] A. Haraux and M. Otani, Analyticity and regularity for a class of second order evolution equations, Evolution Equations and Control Theory, 2 (2013), 101-117. doi: 10.3934/eect.2013.2.101. [22] M. A. Horn and W. Littman, Local smoothing properties of a Schrödinger equation with nonconstant principal part, In: Modelling and optimization of distributed parameter systems, (1996), New York, 104-110. [23] M. A. Horn and W. Littman, Boundary control of a Schrödinger equation with nonconstant principal part, In: Control of partial differential equations and applications, Lecture Notes in Pure and Appl. Math., 174 (1996), New York, 101-106. [24] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [25] W. Littman, The wave operator and $L_p$ norms, J. Math. Mech., 12 (1963), 55-68. [26] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Archive for Rational Mechanics and Analysis, 103 (1988), 193-236. doi: 10.1007/BF00251758. [27] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Annali di Matematica Pura ed Applicata, 152 (1988), 281-330. doi: 10.1007/BF01766154. [28] W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory, J. Anal. Math., 59 (1992), 117-131. doi: 10.1007/BF02790221. [29] W. Littman and S. Taylor, Local smoothing and energy decay for a semi-infinite beam pinned at several points, and applications to boundary control, In: Differential equations, dynamical systems, and control science, Lecture Notes in Pure and Appl. Math., 152 (1994), New York, 683-704. [30] W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), 17-34. doi: 10.1137/S0036139996314106. [31] W. Littman and S. Taylor, The heat and Schrödinger equations: Boundary control with one shot, In: Control Methods in PDE-dynamical Systems, Contemp. Math., 426 (2007), 293-305. doi: 10.1090/conm/426/08194. [32] G. Lumer and R. S. Phillips, On the spectral properties and stabilization of acoustic flow, Pacific J. Math., 11 (1961), 679-698. doi: 10.2140/pjm.1961.11.679. [33] T. Meurer and A. Kugi, Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator, International Journal of Robust and Nonlinear Control, 21 (2011), 542-562. doi: 10.1002/rnc.1611. [34] P. M. Morse and K. U. Ingard, Theoretical Acoustics, Princeton University Press, 1987. [35] D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr., 279 (2006), 299-318. doi: 10.1002/mana.200310362. [36] D. Mugnolo, Damped wave equations with dynamic boundary conditions, J. Appl. Anal., 17 (2011), 241-275. doi: 10.1515/JAA.2011.015. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [38] M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum, 51 (1995), 343-346. doi: 10.1007/BF02573642. [39] S. Taylor, Gevrey's Semigroups, Ph.D. Thesis, University of Minnesota, School of Mathematics 1989. [40] S. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobolev spaces, J. Math. Anal. Appl., 194 (1995), 14-38. doi: 10.1006/jmaa.1995.1284. [41] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second edition. Springer Series in Computational Mathematics, 25. Springer-Verlag, Berlin, 2006. [42] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, In: Operator methods for optimal control problems, Lecture Notes in Pure and Appl. Math., 108 (1987), 283-310. [43] R. P. Vito and S. A. Dixon, Blood Vessel Constitutive Models, Annual Review of Biomedical Engineering, 5 (2003), 413-439. [44] T. J. Xiao and J. Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions, Mathematische Annalen, 327 (2003), 351-363. doi: 10.1007/s00208-003-0457-2. [45] T. J. Xiao and J. Liang, Complete second order differential equations in Banach spaces with dynamic boundary condition, J. Differential Equations, 200 (2004), 105-136. doi: 10.1016/j.jde.2004.01.011. [46] T. J. Xiao and J. Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc., 356 (2004), 4787-4809. doi: 10.1090/S0002-9947-04-03704-3.
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