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Optimal energy decay rates for some wave equations with double damping terms
Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
Department of Mathematics, State University of Maringa, Maringa, 87020-900, Brazil |
$ \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*} $ |
$ \Omega $ |
$ \mathbb{R}^2 $ |
$ \Gamma $ |
$ \omega $ |
$ \partial \Omega $ |
$ \chi_{\omega} $ |
$ \omega $ |
$ a, b\in L^{\infty}(\Omega) $ |
$ a(x) \geq a_0 >0 $ |
$ \omega $ |
$ b(x) \geq b_{0} > 0 $ |
$ \omega $ |
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. |
[2] |
L. Aloui,
Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193.
|
[3] |
L. Aloui,
Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62.
doi: 10.1007/BF03191181. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[17] |
I. Fukuda and M. Tsutsumi,
On coupled Klein-Gordon-Schrödinger equations Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda Univ., 69 (1975), 51-62.
|
[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. |
[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.
|
[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. |
[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.
|
[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. |
[23] |
B. Guo and Y. Li,
Attractor for the dissipative generalized Klein-Gordon-Schrödinger equations, J. Partial Differ. Equations, 11 (1998), 260-272.
|
[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. |
[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. |
[26] |
N. Hayashi,
Global strong solutions of coupled Klein-Gordon-Schrödinger equations, Funkcialaj Ekvacioj, 29 (1986), 299-307.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[32] |
G. Lebeau, Controle de l'equation de Schrödinger. (french) [control of the Schrödinger equation], J. Math. Pures Appl., (9) 71 (1992), 267–291. |
[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. |
[34] |
J. L. Lions, Quelques Métodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Paris, 1969. |
[35] |
J. L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Masson, Paris, 1988. |
[36] |
E. Machtyngier,
Exact controllability for the Schrödinger equation, J. Control and Optimization, 32 (1994), 24-34.
doi: 10.1137/S0363012991223145. |
[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. |
[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. |
[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. |
[40] |
H. Pecher,
Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differ. Int. Equ., 17 (2004), 179-214.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[47] |
H. Yukawa,
On the interaction of elementary particles Ⅰ, Proc. Physico-Math. Soc. Japan, 17 (1935), 48-57.
|
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. |
[2] |
L. Aloui,
Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193.
|
[3] |
L. Aloui,
Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62.
doi: 10.1007/BF03191181. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[17] |
I. Fukuda and M. Tsutsumi,
On coupled Klein-Gordon-Schrödinger equations Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda Univ., 69 (1975), 51-62.
|
[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. |
[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.
|
[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. |
[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.
|
[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. |
[23] |
B. Guo and Y. Li,
Attractor for the dissipative generalized Klein-Gordon-Schrödinger equations, J. Partial Differ. Equations, 11 (1998), 260-272.
|
[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. |
[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. |
[26] |
N. Hayashi,
Global strong solutions of coupled Klein-Gordon-Schrödinger equations, Funkcialaj Ekvacioj, 29 (1986), 299-307.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[32] |
G. Lebeau, Controle de l'equation de Schrödinger. (french) [control of the Schrödinger equation], J. Math. Pures Appl., (9) 71 (1992), 267–291. |
[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. |
[34] |
J. L. Lions, Quelques Métodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Paris, 1969. |
[35] |
J. L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Masson, Paris, 1988. |
[36] |
E. Machtyngier,
Exact controllability for the Schrödinger equation, J. Control and Optimization, 32 (1994), 24-34.
doi: 10.1137/S0363012991223145. |
[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. |
[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. |
[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. |
[40] |
H. Pecher,
Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differ. Int. Equ., 17 (2004), 179-214.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[47] |
H. Yukawa,
On the interaction of elementary particles Ⅰ, Proc. Physico-Math. Soc. Japan, 17 (1935), 48-57.
|
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