Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

Adriana Flores de Almeida; Marcelo Moreira Cavalcanti; Janaina Pedroso Zanchetta

doi:10.3934/eect.2019041

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December 2019, 8(4): 847-865. doi: 10.3934/eect.2019041

Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

Adriana Flores de Almeida , Marcelo Moreira Cavalcanti ^*, and Janaina Pedroso Zanchetta ^**,,

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

^* Corresponding author: janainazanchetta@yahoo.com.br

^* Research partially supported by the CNPq grant 300631/2003-0.
^** Research partially supported by CAPES

Received October 2018 Revised November 2018 Published December 2019 Early access June 2019

The following coupled damped Klein-Gordon-Schrödinger equations are considered

$ \begin{eqnarray*} i\psi_t + \Delta \psi + i \alpha b(x)(|\psi|^{2} + 1)\psi & = & \phi \psi \chi_{\omega} \; \hbox{in}\; \Omega \times (0, \infty), \; (\alpha >0)\ \\ \phi_{tt} - \Delta \phi + a(x) \phi_t & = & |\psi|^2 \chi_{\omega}\; \hbox{in}\; \Omega \times (0, \infty), \end{eqnarray*} $

where

$ \Omega $

is a bounded domain of

$ \mathbb{R}^2 $

, with smooth boundary

$ \Gamma $

and

$ \omega $

is a neighbourhood of

$ \partial \Omega $

satisfying the geometric control condition. Here

$ \chi_{\omega} $

represents the characteristic function of

$ \omega $

. Assuming that

$ a, b\in L^{\infty}(\Omega) $

are nonnegative functions such that

$ a(x) \geq a_0 >0 $

$ \omega $

and

$ b(x) \geq b_{0} > 0 $

$ \omega $

, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et. al in the reference [9] and [1].

Keywords: Klein-Gordon-Schrödinger, localized damping, exponential stability, asymptotic behavior, existence and uniqueness.

Mathematics Subject Classification: Primary: 35L70, 35B40.

Citation: Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041

References:

[1]	A. F. Almeida, M. M. Cavalcanti and J. P. Zanchetta, Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping, Communications on Pure and Applied Analysis, 17 (2018), 2039-2061. doi: 10.3934/cpaa.2018097. Google Scholar
[2]	L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193. Google Scholar
[3]	L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62. doi: 10.1007/BF03191181. Google Scholar
[4]	A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéares, Ann. Inst. H. Poincaré Anal. non Linéaire, 1 (1984), 453-478. doi: 10.1016/S0294-1449(16)30414-0. Google Scholar
[5]	J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Klein-Gordon-Schrödinger equations, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North Holland, Amsterdam, 30 (1978), 37–44. Google Scholar
[6]	C. Banquet, L. C. F. Ferreira and E. J. Villamizar-Roa, On existence and scattering theory for the Klein-Gordon-Schrödinger system in an infinite L2-norm setting, Ann. Mat. Pura Appl., (4) 194 (2015), 781–804. doi: 10.1007/s10231-013-0398-7. Google Scholar
[7]	C. Bardos, G. 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
[8]	P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065. Google Scholar
[9]	V. Bisognin, M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Soriano, Uniform decay for the Klein-Gordon-Schrödinger equations with locally distributed damping, NoDEA, Nonlinear differ. equ. appl., 15 (2008), 91-113. doi: 10.1007/s00030-007-6025-9. Google Scholar
[10]	C. A. Bortot and W. J. Corrêa, Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain, Differential and Integral Equations, 31 (2018), 273-300. Google Scholar
[11]	M. Cavalcanti and V. N. Domingos Cavalcanti, Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations, NoDEA, Nonlinear differ. equ. appl., 7 (2000), 285-307. doi: 10.1007/PL00001426. Google Scholar
[12]	M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction, Journal of Differential Equations, 236 (2007), 407-459. doi: 10.1016/j.jde.2007.02.004. Google Scholar
[13]	J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Soc. Math., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5. Google Scholar
[14]	Z. Dai and P. Gao, Exponential attractor for dissipative Klein-Gordon-Schrödinger equations in ${{\bf{R}}^{\bf{3}}}$, Chim. Ann. Math. Ser. A., 21 (2000), 241-250. Google Scholar
[15]	M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Continuous Dynamical Systems - S, 2 (2009), 67-94. doi: 10.3934/dcdss.2009.2.67. Google Scholar
[16]	B. Dehman, P. 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
[17]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda Univ., 69 (1975), 51-62. Google Scholar
[18]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅱ, J. Math. Analysis Applic., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1. Google Scholar
[19]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon Schrödinger equations Ⅲ - Higher order interaction, decay and blow-up, Math. Japonica, 24 (1979), 307-321. Google Scholar
[20]	I. Fukuda and M. Tsutsumi, On the Yukawa-coupled Klein-Gordon-Schödinger equations in three space dimensions, Proc. Japan Acad., 51 (1975), 402-405. doi: 10.3792/pja/1195518563. Google Scholar
[21]	O. Goubet, A. Hakim and A. Mostafa, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Diff. Integal Equ., 16 (2003), 573-581. Google Scholar
[22]	B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in ${{\bf{R}}^{\bf{3}}}$, Journal of Differential Equations, 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242. Google Scholar
[23]	B. Guo and Y. Li, Attractor for the dissipative generalized Klein-Gordon-Schrödinger equations, J. Partial Differ. Equations, 11 (1998), 260-272. Google Scholar
[24]	B. Guo and Y. Li, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5. Google Scholar
[25]	Y. Han, On the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with rough data, Discret. Contin. Dyn. Syst., 12 (2005), 233-242. doi: 10.3934/dcds.2005.12.233. Google Scholar
[26]	N. Hayashi, Global strong solutions of coupled Klein-Gordon-Schrödinger equations, Funkcialaj Ekvacioj, 29 (1986), 299-307. Google Scholar
[27]	N. Hayashi and W. Von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489. Google Scholar
[28]	H. Lange and B. Wang, Regularity of the global attractor for the Klein-Gordon-Schrödinger equation, Math. Methods Appl. Sci., 22 (1999), 1535-1554. doi: 10.1002/(SICI)1099-1476(19991125)22:17<1535::AID-MMA92>3.0.CO;2-5. Google Scholar
[29]	H. Lange and B. Wang, Attractors for the Klein-Gordon-Schrödinger equation, J. Math. Phys., 40 (1999), 2445-2457. doi: 10.1063/1.532875. Google Scholar
[30]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diffenrential and Integral Equations, 6 (1993), 507-533. Google Scholar
[31]	I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Analysis: Theory, Methods Applications, 64 (2006), 1757-1797. doi: 10.1016/j.na.2005.07.024. Google Scholar
[32]	G. Lebeau, Controle de l'equation de Schrödinger. (french) [control of the Schrödinger equation], J. Math. Pures Appl., (9) 71 (1992), 267–291. Google Scholar
[33]	Y. Li, Q. Shi, C. Wang and S. Wang, Well-posedness for the nonlinear Klein-Gordon-Schrödinger equations with heterointeractions, J. Math. Phys., 51 (2010), 032102, 17pp. doi: 10.1063/1.3317646. Google Scholar
[34]	J. L. Lions, Quelques Métodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Paris, 1969. Google Scholar
[35]	J. L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Masson, Paris, 1988. Google Scholar
[36]	E. Machtyngier, Exact controllability for the Schrödinger equation, J. Control and Optimization, 32 (1994), 24-34. doi: 10.1137/S0363012991223145. Google Scholar
[37]	C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbb{R}^{2+1}$, J. Differ. Equ., 227 (2006), 365-405. doi: 10.1016/j.jde.2005.10.012. Google Scholar
[38]	M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrödinger equations, Nonlinear Analysis, Theory, Methods and Appl., 27 (1996), 455-461. doi: 10.1016/0362-546X(95)00017-P. Google Scholar
[39]	T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23 (1994), 295-305. doi: 10.2969/aspm/02310295. Google Scholar
[40]	H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differ. Int. Equ., 17 (2004), 179-214. Google Scholar
[41]	M. N. Poulou and N. M. Stavrakakis, Global attractor for a system of Klein-Gordon-Schrödinger type in all R, Nonlinear Anal., 74 (2011), 2548-2562. doi: 10.1016/j.na.2010.12.009. Google Scholar
[42]	M. N. Poulou and N. M. Stavrakakis, Uniform decay for a local dissipative Klein-Gordon-Schrödinger type system, Electron. J. Differential Equations, 2012 (2012), 16 pp. Google Scholar
[43]	A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions, J. Math. Sci. Univ. Tokyo, 10 (2003), 661-685. Google Scholar
[44]	A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions. Ⅱ, Hokkaido Math. J., 34 (2005), 405-433. doi: 10.14492/hokmj/1285766230. Google Scholar
[45]	N. Tzirakis, The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space, Comm. Partial Differ. Equ., 30 (2005), 605-641. doi: 10.1081/PDE-200059260. Google Scholar
[46]	B. Wang, Classical global solutions for non-linear Klein-Gordon-Schrödinger equations, Math. Methods Appl. Sci., 20 (1997), 599-616. doi: 10.1002/(SICI)1099-1476(19970510)20:7<599::AID-MMA866>3.0.CO;2-7. Google Scholar
[47]	H. Yukawa, On the interaction of elementary particles Ⅰ, Proc. Physico-Math. Soc. Japan, 17 (1935), 48-57. Google Scholar

show all references

References:

[1]	A. F. Almeida, M. M. Cavalcanti and J. P. Zanchetta, Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping, Communications on Pure and Applied Analysis, 17 (2018), 2039-2061. doi: 10.3934/cpaa.2018097. Google Scholar
[2]	L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193. Google Scholar
[3]	L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62. doi: 10.1007/BF03191181. Google Scholar
[4]	A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéares, Ann. Inst. H. Poincaré Anal. non Linéaire, 1 (1984), 453-478. doi: 10.1016/S0294-1449(16)30414-0. Google Scholar
[5]	J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Klein-Gordon-Schrödinger equations, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North Holland, Amsterdam, 30 (1978), 37–44. Google Scholar
[6]	C. Banquet, L. C. F. Ferreira and E. J. Villamizar-Roa, On existence and scattering theory for the Klein-Gordon-Schrödinger system in an infinite L2-norm setting, Ann. Mat. Pura Appl., (4) 194 (2015), 781–804. doi: 10.1007/s10231-013-0398-7. Google Scholar
[7]	C. Bardos, G. 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
[8]	P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065. Google Scholar
[9]	V. Bisognin, M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Soriano, Uniform decay for the Klein-Gordon-Schrödinger equations with locally distributed damping, NoDEA, Nonlinear differ. equ. appl., 15 (2008), 91-113. doi: 10.1007/s00030-007-6025-9. Google Scholar
[10]	C. A. Bortot and W. J. Corrêa, Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain, Differential and Integral Equations, 31 (2018), 273-300. Google Scholar
[11]	M. Cavalcanti and V. N. Domingos Cavalcanti, Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations, NoDEA, Nonlinear differ. equ. appl., 7 (2000), 285-307. doi: 10.1007/PL00001426. Google Scholar
[12]	M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction, Journal of Differential Equations, 236 (2007), 407-459. doi: 10.1016/j.jde.2007.02.004. Google Scholar
[13]	J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Soc. Math., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5. Google Scholar
[14]	Z. Dai and P. Gao, Exponential attractor for dissipative Klein-Gordon-Schrödinger equations in ${{\bf{R}}^{\bf{3}}}$, Chim. Ann. Math. Ser. A., 21 (2000), 241-250. Google Scholar
[15]	M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Continuous Dynamical Systems - S, 2 (2009), 67-94. doi: 10.3934/dcdss.2009.2.67. Google Scholar
[16]	B. Dehman, P. 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
[17]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda Univ., 69 (1975), 51-62. Google Scholar
[18]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅱ, J. Math. Analysis Applic., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1. Google Scholar
[19]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon Schrödinger equations Ⅲ - Higher order interaction, decay and blow-up, Math. Japonica, 24 (1979), 307-321. Google Scholar
[20]	I. Fukuda and M. Tsutsumi, On the Yukawa-coupled Klein-Gordon-Schödinger equations in three space dimensions, Proc. Japan Acad., 51 (1975), 402-405. doi: 10.3792/pja/1195518563. Google Scholar
[21]	O. Goubet, A. Hakim and A. Mostafa, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Diff. Integal Equ., 16 (2003), 573-581. Google Scholar
[22]	B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in ${{\bf{R}}^{\bf{3}}}$, Journal of Differential Equations, 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242. Google Scholar
[23]	B. Guo and Y. Li, Attractor for the dissipative generalized Klein-Gordon-Schrödinger equations, J. Partial Differ. Equations, 11 (1998), 260-272. Google Scholar
[24]	B. Guo and Y. Li, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5. Google Scholar
[25]	Y. Han, On the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with rough data, Discret. Contin. Dyn. Syst., 12 (2005), 233-242. doi: 10.3934/dcds.2005.12.233. Google Scholar
[26]	N. Hayashi, Global strong solutions of coupled Klein-Gordon-Schrödinger equations, Funkcialaj Ekvacioj, 29 (1986), 299-307. Google Scholar
[27]	N. Hayashi and W. Von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489. Google Scholar
[28]	H. Lange and B. Wang, Regularity of the global attractor for the Klein-Gordon-Schrödinger equation, Math. Methods Appl. Sci., 22 (1999), 1535-1554. doi: 10.1002/(SICI)1099-1476(19991125)22:17<1535::AID-MMA92>3.0.CO;2-5. Google Scholar
[29]	H. Lange and B. Wang, Attractors for the Klein-Gordon-Schrödinger equation, J. Math. Phys., 40 (1999), 2445-2457. doi: 10.1063/1.532875. Google Scholar
[30]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diffenrential and Integral Equations, 6 (1993), 507-533. Google Scholar
[31]	I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Analysis: Theory, Methods Applications, 64 (2006), 1757-1797. doi: 10.1016/j.na.2005.07.024. Google Scholar
[32]	G. Lebeau, Controle de l'equation de Schrödinger. (french) [control of the Schrödinger equation], J. Math. Pures Appl., (9) 71 (1992), 267–291. Google Scholar
[33]	Y. Li, Q. Shi, C. Wang and S. Wang, Well-posedness for the nonlinear Klein-Gordon-Schrödinger equations with heterointeractions, J. Math. Phys., 51 (2010), 032102, 17pp. doi: 10.1063/1.3317646. Google Scholar
[34]	J. L. Lions, Quelques Métodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Paris, 1969. Google Scholar
[35]	J. L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Masson, Paris, 1988. Google Scholar
[36]	E. Machtyngier, Exact controllability for the Schrödinger equation, J. Control and Optimization, 32 (1994), 24-34. doi: 10.1137/S0363012991223145. Google Scholar
[37]	C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbb{R}^{2+1}$, J. Differ. Equ., 227 (2006), 365-405. doi: 10.1016/j.jde.2005.10.012. Google Scholar
[38]	M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrödinger equations, Nonlinear Analysis, Theory, Methods and Appl., 27 (1996), 455-461. doi: 10.1016/0362-546X(95)00017-P. Google Scholar
[39]	T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23 (1994), 295-305. doi: 10.2969/aspm/02310295. Google Scholar
[40]	H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differ. Int. Equ., 17 (2004), 179-214. Google Scholar
[41]	M. N. Poulou and N. M. Stavrakakis, Global attractor for a system of Klein-Gordon-Schrödinger type in all R, Nonlinear Anal., 74 (2011), 2548-2562. doi: 10.1016/j.na.2010.12.009. Google Scholar
[42]	M. N. Poulou and N. M. Stavrakakis, Uniform decay for a local dissipative Klein-Gordon-Schrödinger type system, Electron. J. Differential Equations, 2012 (2012), 16 pp. Google Scholar
[43]	A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions, J. Math. Sci. Univ. Tokyo, 10 (2003), 661-685. Google Scholar
[44]	A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions. Ⅱ, Hokkaido Math. J., 34 (2005), 405-433. doi: 10.14492/hokmj/1285766230. Google Scholar
[45]	N. Tzirakis, The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space, Comm. Partial Differ. Equ., 30 (2005), 605-641. doi: 10.1081/PDE-200059260. Google Scholar
[46]	B. Wang, Classical global solutions for non-linear Klein-Gordon-Schrödinger equations, Math. Methods Appl. Sci., 20 (1997), 599-616. doi: 10.1002/(SICI)1099-1476(19970510)20:7<599::AID-MMA866>3.0.CO;2-7. Google Scholar
[47]	H. Yukawa, On the interaction of elementary particles Ⅰ, Proc. Physico-Math. Soc. Japan, 17 (1935), 48-57. Google Scholar

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2020 Impact Factor: 1.081

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2020 Impact Factor: 1.081

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2020 Impact Factor: 1.081

American Institute of Mathematical Sciences

Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

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American Institute of Mathematical Sciences

Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

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