September  2018, 17(5): 2039-2061. doi: 10.3934/cpaa.2018097

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

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

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

Received  September 2017 Revised  December 2017 Published  April 2018

The following coupled damped Klein-Gordon-Schrödinger equations are considered
$\begin{array}{l}i\psi t + \Delta \psi + i\alpha b(x){( - \Delta )^{\frac{1}{2}}}b(x)\psi = \phi \psi {\chi _\omega }\;{\rm{in}}\;\Omega \times (0,\infty ),(\alpha > 0)\\\phi tt - \Delta \phi + a(x)\phi t = {\left| \psi \right|^2}{\chi _\omega }\;{\rm{in}}\;\Omega \times (0,\infty ),\end{array}$
where
$Ω$
is a bounded domain of
$\mathbb{R}^n$
,
$n = 2$
, with smooth boundary
$Γ$
and
$ω$
is a neighbourhood of
$\partial Ω$
satisfying the geometric control condition. Here
$χ_{ω}$
represents the characteristic function of
$ω$
. Assuming that
$a, b∈ W^{1,∞}(Ω)\cap C^∞(Ω)$
are nonnegative functions such that
$a(x) ≥ a_0 >0$
in
$ω$
and
$b(x) ≥ b_{0} > 0$
in
$ω$
, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous results given by Cavalcanti et. al in the reference [7].
Citation: 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 and Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097
References:
[1]

L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193. 

[2]

L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62. 

[3]

A. Bachelot, Probléme de Cauchy pour des systémes hyperboliques semi-linéares, Ann. Inst. H. Poincaré Anal. non Linéaire, 1 (1984), 453-478. 

[4]

J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Klein-GordonSchr¨odinger equations, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North Holland, Amsterdam, (1978), 37–44.

[5]

C. BanquetL. 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., 194 (2015), 781-804. 

[6]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. 

[7]

V. BisogninM. M. CavalcantiV. 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. 

[8]

C. A. Bortot and M. M. Cavalcanti, Asymptotic stability for the damped Schrödinger equation on noncompact Riemannian manifolds and exterior domains, Comm. Partial Differential Equations, 39 (2014), 1791-1820. 

[9]

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. 

[10]

N. BurqP. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, [Equations de Schrödinger non linéaires dans des domaines extrieurs,], Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 21 (2004), 295-318. 

[11]

M. Cavalcanti and V. Domingos Cavalcanti, Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations, NoDEA, Nonlinear differ. equ. appl., 7 (2000), 285-307. 

[12]

M. M. CavalcantiV. N. Domingos CavalcantiJ. A. Soriano and F. Natali, Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization, J. Differential Equations, 248 (2010), 2955-2971. 

[13]

J. CollianderJ. 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. 

[14]

P. Counstantin and J. C. Saut, Local smoothing properties of dispersive equation, Journal of the American Mathematical Society, 1 (), 413-439. 

[15]

Z. Dai and P. Gao, Exponential attractor for dissipative Klein-Gordon-Schrödinger equations in R3, Chim. Ann. Math. Ser. A., 21 (2000), 241-250. 

[16]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda Univ., 69 (1975), 51-62. 

[17]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅱ, J. Math. Analysis Applic., 66 (1978), 358-378. 

[18]

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. 

[19]

I. Fukuda and M. Tsutsumi, On the Yukawa-coupled Klein-Gordon-Schödinger equations in three space dimensions, Proc. Japan Acad., 51 (1975), 402-405. 

[20]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\bf{R}^3$, Journal of Differential Equations, 136 (1997), 356-377. 

[21]

B. Guo and Y. Li, Attractor for the dissipative generalized Klein-Gordon-Schrödinger equations, J. Partial Differ. Equations, 11 (1998), 260-272. 

[22]

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. 

[23]

Y. Han, On the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with rough data, Discret. Contin. Dyn. Syst., 12 (2005), 233-242. 

[24]

N. Hayashi, Global strong solutions of coupled Klein-Gordon-Schrödinger equations, Funkcialaj Ekvacioj, 29 (1986), 299-307. 

[25]

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. 

[26]

H. Lange and B. Wang, Regularity of the global attractor for the Klein-Gordon-Schrödinger equation, Math. Methods Appl. Sci., 22 (1999), 1535-1554. 

[27]

H. Lange and B. Wang, Attractors for the Klein-Gordon-Schrödinger equation, J. Math. Phys., 40 (1999), 2445-2457. 

[28]

Y. LiQ. ShiC. Wang and S. Wang, Well-posedness for the nonlinear Klein-Gordon-Schrödinger equations with heterointeractions, J. Math. Phys., 51 (2010), 032-102. 

[29]

J. L. Lions, Quelques Métodes de Résolution des Problémes Aux Limites Non Linéaires, Dunod, Paris, 1969.

[30]

J. L. Lions, Controlabilité exacte, perturbations et Stabilisation de systèmes distribués, Tome 1, Masson, Paris, 1988.

[31]

J. L. Lions-E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Dunod, Paris, 1968, Vol. 1.

[32]

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. 

[33]

O. GoubetA. Hakim and A. Mostafa, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Diff. Integal Equ., 16 (2003), 573-581. 

[34]

M. Ohta, Stability of Stationary States for the Coupled Klein-Gordon-Schrödinger Equations, Nonlinear Analysis, Theory, Methods and Appl., 27 (1996), 455-461. 

[35]

T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23 (1994), 295-305. 

[36]

H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differ. Int. Equ., 17 (2004), 179-214. 

[37]

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. 

[38]

M. N. Poulou and N. M. Stavrakakis, Uniform decay for a local dissipative Klein-Gordon-Schrödinger type system, Electron. J. Differential Equations, (2012), No. 179. 16pp.

[39]

A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions, J. Math. Sci. Univ. Tokyo, 10 (2003), 661-685. 

[40]

A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions. Ⅱ, Hokkaido Math. J., 34 (2005), 405-433. 

[41]

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. 

[42]

B. Wang, Classical global solutions for non-linear Klein-Gordon-Schrödinger equations, Math. Methods Appl. Sci., 20 (1997), 599-616. 

[43]

H. Yukawa, On the interaction of elementary particles Ⅰ, Proc. Physico-Math. Soc. Japan, 17 (1935), 48-57. 

show all references

References:
[1]

L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193. 

[2]

L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62. 

[3]

A. Bachelot, Probléme de Cauchy pour des systémes hyperboliques semi-linéares, Ann. Inst. H. Poincaré Anal. non Linéaire, 1 (1984), 453-478. 

[4]

J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Klein-GordonSchr¨odinger equations, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North Holland, Amsterdam, (1978), 37–44.

[5]

C. BanquetL. 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., 194 (2015), 781-804. 

[6]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. 

[7]

V. BisogninM. M. CavalcantiV. 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. 

[8]

C. A. Bortot and M. M. Cavalcanti, Asymptotic stability for the damped Schrödinger equation on noncompact Riemannian manifolds and exterior domains, Comm. Partial Differential Equations, 39 (2014), 1791-1820. 

[9]

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. 

[10]

N. BurqP. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, [Equations de Schrödinger non linéaires dans des domaines extrieurs,], Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 21 (2004), 295-318. 

[11]

M. Cavalcanti and V. Domingos Cavalcanti, Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations, NoDEA, Nonlinear differ. equ. appl., 7 (2000), 285-307. 

[12]

M. M. CavalcantiV. N. Domingos CavalcantiJ. A. Soriano and F. Natali, Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization, J. Differential Equations, 248 (2010), 2955-2971. 

[13]

J. CollianderJ. 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. 

[14]

P. Counstantin and J. C. Saut, Local smoothing properties of dispersive equation, Journal of the American Mathematical Society, 1 (), 413-439. 

[15]

Z. Dai and P. Gao, Exponential attractor for dissipative Klein-Gordon-Schrödinger equations in R3, Chim. Ann. Math. Ser. A., 21 (2000), 241-250. 

[16]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda Univ., 69 (1975), 51-62. 

[17]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅱ, J. Math. Analysis Applic., 66 (1978), 358-378. 

[18]

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. 

[19]

I. Fukuda and M. Tsutsumi, On the Yukawa-coupled Klein-Gordon-Schödinger equations in three space dimensions, Proc. Japan Acad., 51 (1975), 402-405. 

[20]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\bf{R}^3$, Journal of Differential Equations, 136 (1997), 356-377. 

[21]

B. Guo and Y. Li, Attractor for the dissipative generalized Klein-Gordon-Schrödinger equations, J. Partial Differ. Equations, 11 (1998), 260-272. 

[22]

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. 

[23]

Y. Han, On the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with rough data, Discret. Contin. Dyn. Syst., 12 (2005), 233-242. 

[24]

N. Hayashi, Global strong solutions of coupled Klein-Gordon-Schrödinger equations, Funkcialaj Ekvacioj, 29 (1986), 299-307. 

[25]

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. 

[26]

H. Lange and B. Wang, Regularity of the global attractor for the Klein-Gordon-Schrödinger equation, Math. Methods Appl. Sci., 22 (1999), 1535-1554. 

[27]

H. Lange and B. Wang, Attractors for the Klein-Gordon-Schrödinger equation, J. Math. Phys., 40 (1999), 2445-2457. 

[28]

Y. LiQ. ShiC. Wang and S. Wang, Well-posedness for the nonlinear Klein-Gordon-Schrödinger equations with heterointeractions, J. Math. Phys., 51 (2010), 032-102. 

[29]

J. L. Lions, Quelques Métodes de Résolution des Problémes Aux Limites Non Linéaires, Dunod, Paris, 1969.

[30]

J. L. Lions, Controlabilité exacte, perturbations et Stabilisation de systèmes distribués, Tome 1, Masson, Paris, 1988.

[31]

J. L. Lions-E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Dunod, Paris, 1968, Vol. 1.

[32]

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. 

[33]

O. GoubetA. Hakim and A. Mostafa, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Diff. Integal Equ., 16 (2003), 573-581. 

[34]

M. Ohta, Stability of Stationary States for the Coupled Klein-Gordon-Schrödinger Equations, Nonlinear Analysis, Theory, Methods and Appl., 27 (1996), 455-461. 

[35]

T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23 (1994), 295-305. 

[36]

H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differ. Int. Equ., 17 (2004), 179-214. 

[37]

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. 

[38]

M. N. Poulou and N. M. Stavrakakis, Uniform decay for a local dissipative Klein-Gordon-Schrödinger type system, Electron. J. Differential Equations, (2012), No. 179. 16pp.

[39]

A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions, J. Math. Sci. Univ. Tokyo, 10 (2003), 661-685. 

[40]

A. Shimomura, Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions. Ⅱ, Hokkaido Math. J., 34 (2005), 405-433. 

[41]

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. 

[42]

B. Wang, Classical global solutions for non-linear Klein-Gordon-Schrödinger equations, Math. Methods Appl. Sci., 20 (1997), 599-616. 

[43]

H. Yukawa, On the interaction of elementary particles Ⅰ, Proc. Physico-Math. Soc. Japan, 17 (1935), 48-57. 

Figure 1.  Example of a region where Conjecture 2 is satisfied.
Figure 2.  Neighborhood $\hat{\omega}$ of $\overline{\Gamma(x^0)}$
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