# American Institute of Mathematical Sciences

December  2015, 5(4): 761-780. doi: 10.3934/mcrf.2015.5.761

## Stabilization of hyperbolic equations with mixed boundary conditions

 1 School of Mathematics, Sichuan University, Chengdu 610064, China

Received  July 2014 Revised  January 2015 Published  October 2015

This paper is devoted to study decay properties of solutions to hyperbolic equations in a bounded domain with two types of dissipative mechanisms, i.e. either with a small boundary or an internal damping. Both of the equations are equipped with the mixed boundary conditions. When the Geometric Control Condition on the dissipative region is not satisfied, we show that sufficiently smooth solutions to the equations decay logarithmically, under sharp regularity assumptions on the coefficients, the damping and the boundary of the domain involved in the equations. Our decay results rely on an analysis of the size of resolvent operators for hyperbolic equations on the imaginary axis. To derive this kind of resolvent estimates, we employ global Carleman estimates for elliptic equations with mixed boundary conditions.
Citation: Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761
##### References:
 [1] K. Ammari, Dirichlet boundary stabilization of the wave equation,, Asymptot. Anal., 30 (2002), 117.   Google Scholar [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar [3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765.  doi: 10.1007/s00028-008-0424-1.  Google Scholar [4] M. Bellassoued, Decay of solutions of the elastic wave equation with a localized dissipation,, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 267.  doi: 10.5802/afst.1049.  Google Scholar [5] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar [6] N. 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Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping,, in Some Problems on Nonlinear Hyperbolic Equations and Applications, (2010), 310.  doi: 10.1080/03605300903116389.  Google Scholar [11] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping,, Z. angew. Math. Phys., 62 (2011), 667.  doi: 10.1007/s00033-010-0113-0.  Google Scholar [12] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, (French) [Dissipative Dynamical Systems and Applications], (1991).   Google Scholar [13] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar [14] O. Yu. Imanuvilov and M. Yamomoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate,, Inverse Problems, 14 (1998), 1229.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar [15] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33.   Google Scholar [16] G. Lebeau, Equation des ondes amorties,, (French) [Damped wave equation], (1993), 73.   Google Scholar [17] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord,, (French) [Stabilization of the wave equations by the boundary], 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar [18] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar [19] Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators,, ESAIM Control Optim. Calc. Var., 19 (2013), 255.  doi: 10.1051/cocv/2012008.  Google Scholar [20] K.-D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar [21] K.-D. Phung and X. Zhang, Time reversal focusing of the initial state for Kirchoff plate,, SIAM J. Appl. Math., 68 (2008), 1535.  doi: 10.1137/070684823.  Google Scholar [22] J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [23] T. Qin, The global smooth solutions of second order quasilinear hyperbolic equations with dissipative boundary conditions,, Chinese Ann. Math. Ser. B, 9 (1988), 251.   Google Scholar [24] J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds,, Comm. Pure Appl. Math., 28 (1975), 501.  doi: 10.1002/cpa.3160280405.  Google Scholar [25] M. Reed and B. Simon, Methods of Modern Mathematics Physics II: Fourier Analysis, Self-Adjointness,, Academic Press, (1975).   Google Scholar [26] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations,, in Proc. Internat. Congress of Mathematicians, (1971), 3008.  doi: 10.1007/978-0-387-89488-1.  Google Scholar [27] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar

show all references

##### References:
 [1] K. Ammari, Dirichlet boundary stabilization of the wave equation,, Asymptot. Anal., 30 (2002), 117.   Google Scholar [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar [3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765.  doi: 10.1007/s00028-008-0424-1.  Google Scholar [4] M. Bellassoued, Decay of solutions of the elastic wave equation with a localized dissipation,, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 267.  doi: 10.5802/afst.1049.  Google Scholar [5] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar [6] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel,, (French) [Decay of the local energy of the wave equation for the exterior problem and absence of resonance near the real axis], 180 (1998), 1.  doi: 10.1007/BF02392877.  Google Scholar [7] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.  doi: 10.4310/MRL.2007.v14.n1.a3.  Google Scholar [8] H. Christianson, E. Schenck, A. Vasy and J. Wunsch, From resolvent estimates to damped waves,, J. Anal. Math., 122 (2014), 14.  doi: 10.1007/s11854-014-0006-9.  Google Scholar [9] P. Cornilleau and L. Robbiano, Carleman estimates for the Zaremba boundary condition and stabilization of waves,, Amer. J. Math., 136 (2014), 393.  doi: 10.1353/ajm.2014.0014.  Google Scholar [10] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping,, in Some Problems on Nonlinear Hyperbolic Equations and Applications, (2010), 310.  doi: 10.1080/03605300903116389.  Google Scholar [11] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping,, Z. angew. Math. Phys., 62 (2011), 667.  doi: 10.1007/s00033-010-0113-0.  Google Scholar [12] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, (French) [Dissipative Dynamical Systems and Applications], (1991).   Google Scholar [13] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar [14] O. Yu. Imanuvilov and M. Yamomoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate,, Inverse Problems, 14 (1998), 1229.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar [15] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33.   Google Scholar [16] G. Lebeau, Equation des ondes amorties,, (French) [Damped wave equation], (1993), 73.   Google Scholar [17] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord,, (French) [Stabilization of the wave equations by the boundary], 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar [18] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar [19] Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators,, ESAIM Control Optim. Calc. Var., 19 (2013), 255.  doi: 10.1051/cocv/2012008.  Google Scholar [20] K.-D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar [21] K.-D. Phung and X. Zhang, Time reversal focusing of the initial state for Kirchoff plate,, SIAM J. Appl. Math., 68 (2008), 1535.  doi: 10.1137/070684823.  Google Scholar [22] J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [23] T. Qin, The global smooth solutions of second order quasilinear hyperbolic equations with dissipative boundary conditions,, Chinese Ann. Math. Ser. B, 9 (1988), 251.   Google Scholar [24] J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds,, Comm. Pure Appl. Math., 28 (1975), 501.  doi: 10.1002/cpa.3160280405.  Google Scholar [25] M. Reed and B. Simon, Methods of Modern Mathematics Physics II: Fourier Analysis, Self-Adjointness,, Academic Press, (1975).   Google Scholar [26] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations,, in Proc. Internat. Congress of Mathematicians, (1971), 3008.  doi: 10.1007/978-0-387-89488-1.  Google Scholar [27] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar
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