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

Previous Article
Existence and extinction in finite time for Stratonovich gradient noise porous media equations
EECT Home
This Issue
Next Article
Optimal energy decay rates for some wave equations with double damping terms

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

Figure 1.

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55
[2]	A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097
[3]	Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221
[4]	Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413
[5]	Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149
[6]	Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4021-4044. doi: 10.3934/dcdsb.2018122
[7]	Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
[8]	Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[9]	Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695
[10]	Ahmed Y. Abdallah, Taqwa M. Al-Khader, Heba N. Abu-Shaab. Attractors of the Klein-Gordon-Schrödinger lattice systems with almost periodic nonlinear part. Discrete & Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022006
[11]	Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239
[12]	E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
[13]	Salah Missaoui. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system. Communications on Pure & Applied Analysis, 2022, 21 (2) : 567-584. doi: 10.3934/cpaa.2021189
[14]	Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[15]	Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[16]	Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[17]	Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233
[18]	P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029
[19]	Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795
[20]	Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

2020 Impact Factor: 1.081

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.081

Tools

Figures and Tables

2020 Impact Factor: 1.081

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors