September  2016, 15(5): 1743-1756. doi: 10.3934/cpaa.2016011

Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$

1. 

LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex, France

2. 

LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex

Received  September 2015 Revised  March 2016 Published  July 2016

We consider a damped forced nonlinear Schrödinger-Poisson system in $R^3$. This provides us with an infinite-dimensional dynamical system in the energy space $H^1(R^3)$. We prove the existence of a finite dimensional global attractor for this dynamical system.
Citation: Amna Dabaa, O. Goubet. Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1743-1756. doi: 10.3934/cpaa.2016011
References:
[1]

N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$, Appl. Math. Lett., 12 (1999), 45-48. doi: 10.1016/S0893-9659(98)00170-0.  Google Scholar

[2]

J. Ball, Global attractors for damped semilinear wave equations, Partial Differential Equations and Applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.  Google Scholar

[3]

K. Benmlih, Stationary solutions for schrödinger-Poisson system in $\mathbbR^3$, Proceedings of the 2002 Fez Conference on Partial Differential Equations, 65-76 (electronic), Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002.  Google Scholar

[4]

K. Benmlih and O. Kavian, Existence and asymptotic behaviour of standing waves for quasilinear Schröinger-Poisson systems in $R^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 449-470.  Google Scholar

[5]

B. Bongioanni et J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360. doi: 10.1007/BF02829750.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar

[8]

T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Ellipses, Paris, 1990.  Google Scholar

[9]

A. Cipolatti and O. Kavian, On a nonlinear Schröinger equation modelling ultra-short laser pulses with a large noncompact global attractor, Discrete Contin. Dyn. Syst., 17 (2007), 121-132.  Google Scholar

[10]

A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson, Thèse, Université de Picardie Jules Verne, 2010. Google Scholar

[11]

A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson sur un domaine borné de $\mathbbR^3$, Ann. Math. Blaise Pascal, 17 (2010), 199-232.  Google Scholar

[12]

J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations, Ann.Inst. Henri Poincaré., 5 (1988), 365-405.  Google Scholar

[13]

J.-M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schröinger equations. Attractors, inertial manifolds and their approximation, RAIRO Modé. Math. Anal. Numé, 23 (1989), 433-443.  Google Scholar

[14]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420.  Google Scholar

[15]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$, Adv. Differential Equations, 3 (1998), 337-360.  Google Scholar

[16]

O. Goubet and M. Hussein, Global attractor for the Davey-Stewartson system on $R^2$, Commun. Pure Appl. Anal., 8 (2009), 1555-1575.  Google Scholar

[17]

O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schröinger equations in $L^2(R)$, Nonlinear Anal., 71 (2009), 317-320.  Google Scholar

[18]

J. Hale, Asymptotic Behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence.  Google Scholar

[19]

R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems, J. Differential equations, 145 (1998), 1-16.  Google Scholar

[20]

R. Illner, H. Lange, B. Toomir and P. Zweifel, On quasi-linear Schrödinger-Poisson systems, Math. Meth. Appl. Sci., 20 (1997), 1223-1238.  Google Scholar

[21]

M. Jolly, T. Sadigov and E. Titi, A determining form for the damped driven nonlinear Schröinger equationfourier modes case, J. Differential Equations, 258 (2015), 2711-2744.  Google Scholar

[22]

P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369. doi: 10.1007/BF01261181.  Google Scholar

[23]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990.  Google Scholar

[24]

A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 10300, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[25]

L. Molinet, Global attractor and asymptotic smoothing effects for the weakly damped cubic Schröinger equation in $L^2(T)$, Dyn. Partial Differ. Equ., 6 (2009), 154.  Google Scholar

[26]

F. Nier, Schrödinger-Poisson systems in dimension $d\leq3$, the whole space case, Proceedings of the Royal Society of Edinburgh, 123A (1993), 1179-1201. Google Scholar

[27]

M. Oliver and E. Titi, Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schröinger equation, Indiana Univ. Math. J., 47 (1998), 493.  Google Scholar

[28]

G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[29]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997.  Google Scholar

[30]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors, Physica D, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

show all references

References:
[1]

N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$, Appl. Math. Lett., 12 (1999), 45-48. doi: 10.1016/S0893-9659(98)00170-0.  Google Scholar

[2]

J. Ball, Global attractors for damped semilinear wave equations, Partial Differential Equations and Applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.  Google Scholar

[3]

K. Benmlih, Stationary solutions for schrödinger-Poisson system in $\mathbbR^3$, Proceedings of the 2002 Fez Conference on Partial Differential Equations, 65-76 (electronic), Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002.  Google Scholar

[4]

K. Benmlih and O. Kavian, Existence and asymptotic behaviour of standing waves for quasilinear Schröinger-Poisson systems in $R^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 449-470.  Google Scholar

[5]

B. Bongioanni et J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360. doi: 10.1007/BF02829750.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar

[8]

T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Ellipses, Paris, 1990.  Google Scholar

[9]

A. Cipolatti and O. Kavian, On a nonlinear Schröinger equation modelling ultra-short laser pulses with a large noncompact global attractor, Discrete Contin. Dyn. Syst., 17 (2007), 121-132.  Google Scholar

[10]

A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson, Thèse, Université de Picardie Jules Verne, 2010. Google Scholar

[11]

A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson sur un domaine borné de $\mathbbR^3$, Ann. Math. Blaise Pascal, 17 (2010), 199-232.  Google Scholar

[12]

J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations, Ann.Inst. Henri Poincaré., 5 (1988), 365-405.  Google Scholar

[13]

J.-M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schröinger equations. Attractors, inertial manifolds and their approximation, RAIRO Modé. Math. Anal. Numé, 23 (1989), 433-443.  Google Scholar

[14]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420.  Google Scholar

[15]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$, Adv. Differential Equations, 3 (1998), 337-360.  Google Scholar

[16]

O. Goubet and M. Hussein, Global attractor for the Davey-Stewartson system on $R^2$, Commun. Pure Appl. Anal., 8 (2009), 1555-1575.  Google Scholar

[17]

O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schröinger equations in $L^2(R)$, Nonlinear Anal., 71 (2009), 317-320.  Google Scholar

[18]

J. Hale, Asymptotic Behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence.  Google Scholar

[19]

R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems, J. Differential equations, 145 (1998), 1-16.  Google Scholar

[20]

R. Illner, H. Lange, B. Toomir and P. Zweifel, On quasi-linear Schrödinger-Poisson systems, Math. Meth. Appl. Sci., 20 (1997), 1223-1238.  Google Scholar

[21]

M. Jolly, T. Sadigov and E. Titi, A determining form for the damped driven nonlinear Schröinger equationfourier modes case, J. Differential Equations, 258 (2015), 2711-2744.  Google Scholar

[22]

P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369. doi: 10.1007/BF01261181.  Google Scholar

[23]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990.  Google Scholar

[24]

A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 10300, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[25]

L. Molinet, Global attractor and asymptotic smoothing effects for the weakly damped cubic Schröinger equation in $L^2(T)$, Dyn. Partial Differ. Equ., 6 (2009), 154.  Google Scholar

[26]

F. Nier, Schrödinger-Poisson systems in dimension $d\leq3$, the whole space case, Proceedings of the Royal Society of Edinburgh, 123A (1993), 1179-1201. Google Scholar

[27]

M. Oliver and E. Titi, Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schröinger equation, Indiana Univ. Math. J., 47 (1998), 493.  Google Scholar

[28]

G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[29]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997.  Google Scholar

[30]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors, Physica D, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

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