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November  2020, 19(11): 5131-5156. doi: 10.3934/cpaa.2020230

Uniform stabilization of the Klein-Gordon system

1. 

Department of Mathematics, State University of Maringá, 87020-900, Maringá, Brazil

2. 

Department of Mathematics, Federal University of Pampa - Campus Itaqui, 97650-000, Itaqui, Brazil

3. 

Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

* Corresponding author

Received  October 2019 Revised  June 2020 Published  November 2020 Early access  September 2020

Fund Project: Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0. Research of Victor Hugo Gonzalez Martinez is partially supported by CAPES

We consider the Klein-Gordon system posed in an inhomogeneous medium $ \Omega $ with smooth boundary $ \partial \Omega $ subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood $ \omega $ of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [5]. Although the present problem has some similarity to the reference [6] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.

Citation: Marcelo M. Cavalcanti, Leonel G. Delatorre, Daiane C. Soares, Victor Hugo Gonzalez Martinez, Janaina P. Zanchetta. Uniform stabilization of the Klein-Gordon system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5131-5156. doi: 10.3934/cpaa.2020230
References:
[1]

D. Andrade and A. Mognon, Global solutions for a system of Klein-Gordon Equations with memory, Bol. Soc. Parana. Mat., 3 (2003), 127-138.  doi: 10.5269/bspm.v21i1-2.7512.

[2]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[3]

S. BetelúR. Gulliver and and W. Littman., Boundary control of PDEs via curvature flows: the view from the boundary. II., Appl. Math. Optim., 46 (2002), 67-178.  doi: 10.1007/s00245-002-0742-6.

[4]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752.  doi: 10.1016/S0764-4442(97)80053-5.

[5]

N. Burq and P. Gérard, Contrôle Optimal des équations aux dérivées partielles, P. Contrôle Optimal des équations aux dérivées partielles. 2001

[6]

M. M. Cavalcanti et al., Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping, J. Differ. Equ., 268 (2020), 447–489. doi: 10.1016/j.jde.2019.08.011.

[7]

M. M. Cavalcanti and V. N. Domingos Cavalcanti, Introdução à teoria das distribuições e aos espaços de Sobolev, Editora da Universidade Estadual de Maringá (Eduem), Maringá, 2009.

[8]

M. M. CavalcantiV. N. Domingos Cavalcanti and R. Fukuoka, Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result, Trans. Amer. Math. Soc., 361 (2009), 4561-4580.  doi: 10.1090/S0002-9947-09-04763-1.

[9]

M. M. CavalcantiV. N. Domingos CavalcantiR. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Arch. Ration. Mech. Anal., 197 (2010), 925-964.  doi: 10.1007/s00205-009-0284-z.

[10]

M. M. Cavalcanti, V. N. D. Cavalcanti and V. Komornik, Introdução a Análise Funcional, Editora da Universidade Estadual de Maringá (Eduem), Maringá, 2011.

[11]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. S. Prates Filho, Existence and uniform decay of a degenerate and generalized Klein-Gordon system with boundary dampin, Commun. Appl. Anal., 4 (2000), 173-196. 

[12]

A. T. CousinC. L. Frota and N. A. Larkin, On a system of Klein-Gordon type equations with acoustic boundary conditions, J. Math. Anal. Appl., 293 (2004), 293-309.  doi: 10.1016/j.jmaa.2004.01.007.

[13]

B. DehmanP. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.

[14]

B. DehmanG. Lebeau and and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36 (2003), 525-551.  doi: 10.1016/S0012-9593(03)00021-1.

[15]

D. Dos Santos Ferreira, Sharp $L^p$ Carleman estimates and unique continuation, Journées "Équations aux Dérivées Partielles", pages Exp. No. VI, 12. Univ. Nantes, Nantes, 2003.

[16]

J. S. Ferreira, Asymptotic behavior of the solutions of a nonlinear system of Klein-Gordon equations, Nonlinear Anal., 13 (1989), 1115-1126.  doi: 10.1016/0362-546X(89)90098-9.

[17]

J. S. Ferreira, Exponential decay for a nonlinear system of hyperbolic equations with locally distributed dampings, Nonlinear Anal., 18 (1992), 1015-1032.  doi: 10.1016/0362-546X(92)90193-I.

[18]

J. S. Ferreira, Exponential decay of the energy of a nonlinear system of Klein-Gordon equations with localized dampings in bounded and unbounded domains, Asymptotic Anal., 8 (1994), 73-92. 

[19]

J. Ferreira and and G. P. Menzala, Decay of solutions of a system of nonlinear Klein-Gordon equations, Internat. J. Math. Math. Sci., 9 (1986), 471-483.  doi: 10.1155/S0161171286000601.

[20]

P. Gérard, Microlocal defect measures, Commun. Partial. Differ. Equ., 16 (1991), 1761-1794.  doi: 10.1080/03605309108820822.

[21]

J. Jost, Riemannian Geometry and Geometric Analysis, Springer Verlag, 2008.

[22]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.

[23]

J. L. Lions, Quelques méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Guthier-Villars, 1969.

[24]

K. Liu and K. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control and Optim. 36 (1998), 1086–1098. doi: 10.1137/S0363012996310703.

[25]

K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419–432. doi: 10.1007/s00033-005-0029-2.

[26]

L. A. Medeiros and G. P. Menzala, On a mixed problem for a class of nonlinear Klein-Gordon equations, Acta Math. Hungar., 52 (1988), 61-69.  doi: 10.1007/BF01952481.

[27]

L. A. Medeiros and M. M. Miranda, Weak solutions for a system of nonlinear Klein-Gordon equations, Ann. Mat. Pura Appl., 146 (1987), 173-183.  doi: 10.1007/BF01762364.

[28]

L. A. Medeiros and M. M. Miranda, On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkcial. Ekvac., 30 (1987), 147-161. 

[29]

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.

[30]

J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math., 28 (1975), 501-523.  doi: 10.1002/cpa.3160280405.

[31]

L. Robbiano and Q. Zhang, Logarithmic Decay of a Wave Equation with Kelvin-Voigt Damping, Preprint arXiv: 1809.03196.

[32]

A. Ruiz, Unique Continuation for Weak Solutions of the Wave Equation plus a Potential, J. Math. Pures. Appl., 71 (1992), 455-467. 

[33]

I. E. Segal, Nonlinear partial differential equations in quantum field theory, Proc. Sympos. Appl. Math., Vol. XVII, 1965,210–226.

[34]

J. Simon, Compact sets in the space $L^p(0, T, B).$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[35]

L. Tebou, Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping, Discrete Cont. Dyn-A, 36 (2016), 7117-7136.  doi: 10.3934/dcds.2016110.

show all references

References:
[1]

D. Andrade and A. Mognon, Global solutions for a system of Klein-Gordon Equations with memory, Bol. Soc. Parana. Mat., 3 (2003), 127-138.  doi: 10.5269/bspm.v21i1-2.7512.

[2]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[3]

S. BetelúR. Gulliver and and W. Littman., Boundary control of PDEs via curvature flows: the view from the boundary. II., Appl. Math. Optim., 46 (2002), 67-178.  doi: 10.1007/s00245-002-0742-6.

[4]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752.  doi: 10.1016/S0764-4442(97)80053-5.

[5]

N. Burq and P. Gérard, Contrôle Optimal des équations aux dérivées partielles, P. Contrôle Optimal des équations aux dérivées partielles. 2001

[6]

M. M. Cavalcanti et al., Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping, J. Differ. Equ., 268 (2020), 447–489. doi: 10.1016/j.jde.2019.08.011.

[7]

M. M. Cavalcanti and V. N. Domingos Cavalcanti, Introdução à teoria das distribuições e aos espaços de Sobolev, Editora da Universidade Estadual de Maringá (Eduem), Maringá, 2009.

[8]

M. M. CavalcantiV. N. Domingos Cavalcanti and R. Fukuoka, Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result, Trans. Amer. Math. Soc., 361 (2009), 4561-4580.  doi: 10.1090/S0002-9947-09-04763-1.

[9]

M. M. CavalcantiV. N. Domingos CavalcantiR. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Arch. Ration. Mech. Anal., 197 (2010), 925-964.  doi: 10.1007/s00205-009-0284-z.

[10]

M. M. Cavalcanti, V. N. D. Cavalcanti and V. Komornik, Introdução a Análise Funcional, Editora da Universidade Estadual de Maringá (Eduem), Maringá, 2011.

[11]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. S. Prates Filho, Existence and uniform decay of a degenerate and generalized Klein-Gordon system with boundary dampin, Commun. Appl. Anal., 4 (2000), 173-196. 

[12]

A. T. CousinC. L. Frota and N. A. Larkin, On a system of Klein-Gordon type equations with acoustic boundary conditions, J. Math. Anal. Appl., 293 (2004), 293-309.  doi: 10.1016/j.jmaa.2004.01.007.

[13]

B. DehmanP. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.

[14]

B. DehmanG. Lebeau and and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36 (2003), 525-551.  doi: 10.1016/S0012-9593(03)00021-1.

[15]

D. Dos Santos Ferreira, Sharp $L^p$ Carleman estimates and unique continuation, Journées "Équations aux Dérivées Partielles", pages Exp. No. VI, 12. Univ. Nantes, Nantes, 2003.

[16]

J. S. Ferreira, Asymptotic behavior of the solutions of a nonlinear system of Klein-Gordon equations, Nonlinear Anal., 13 (1989), 1115-1126.  doi: 10.1016/0362-546X(89)90098-9.

[17]

J. S. Ferreira, Exponential decay for a nonlinear system of hyperbolic equations with locally distributed dampings, Nonlinear Anal., 18 (1992), 1015-1032.  doi: 10.1016/0362-546X(92)90193-I.

[18]

J. S. Ferreira, Exponential decay of the energy of a nonlinear system of Klein-Gordon equations with localized dampings in bounded and unbounded domains, Asymptotic Anal., 8 (1994), 73-92. 

[19]

J. Ferreira and and G. P. Menzala, Decay of solutions of a system of nonlinear Klein-Gordon equations, Internat. J. Math. Math. Sci., 9 (1986), 471-483.  doi: 10.1155/S0161171286000601.

[20]

P. Gérard, Microlocal defect measures, Commun. Partial. Differ. Equ., 16 (1991), 1761-1794.  doi: 10.1080/03605309108820822.

[21]

J. Jost, Riemannian Geometry and Geometric Analysis, Springer Verlag, 2008.

[22]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.

[23]

J. L. Lions, Quelques méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Guthier-Villars, 1969.

[24]

K. Liu and K. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control and Optim. 36 (1998), 1086–1098. doi: 10.1137/S0363012996310703.

[25]

K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419–432. doi: 10.1007/s00033-005-0029-2.

[26]

L. A. Medeiros and G. P. Menzala, On a mixed problem for a class of nonlinear Klein-Gordon equations, Acta Math. Hungar., 52 (1988), 61-69.  doi: 10.1007/BF01952481.

[27]

L. A. Medeiros and M. M. Miranda, Weak solutions for a system of nonlinear Klein-Gordon equations, Ann. Mat. Pura Appl., 146 (1987), 173-183.  doi: 10.1007/BF01762364.

[28]

L. A. Medeiros and M. M. Miranda, On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkcial. Ekvac., 30 (1987), 147-161. 

[29]

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.

[30]

J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math., 28 (1975), 501-523.  doi: 10.1002/cpa.3160280405.

[31]

L. Robbiano and Q. Zhang, Logarithmic Decay of a Wave Equation with Kelvin-Voigt Damping, Preprint arXiv: 1809.03196.

[32]

A. Ruiz, Unique Continuation for Weak Solutions of the Wave Equation plus a Potential, J. Math. Pures. Appl., 71 (1992), 455-467. 

[33]

I. E. Segal, Nonlinear partial differential equations in quantum field theory, Proc. Sympos. Appl. Math., Vol. XVII, 1965,210–226.

[34]

J. Simon, Compact sets in the space $L^p(0, T, B).$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[35]

L. Tebou, Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping, Discrete Cont. Dyn-A, 36 (2016), 7117-7136.  doi: 10.3934/dcds.2016110.

Figure 1.  The Kelvin-Voigt dampings act in $O_1 = \Omega \backslash A$ and $O_2 = \Omega \backslash B$ while the frictional dampings are effective in a collar of $\partial A$ and $\partial B$
Figure 2.  Admissible geometries for the Kelvin-Voigt and frictional dissipations $ a(x) $ and $ \gamma_1(x) $, respectively
Figure 3.  Admissible geometries for the Kelvin-Voigt and frictional dissipations $ a(x) $ and $ \gamma_1(x) $, respectively
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