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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

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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