# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [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. [8] T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Ellipses, Paris, 1990. [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. [10] A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson, Thèse, Université de Picardie Jules Verne, 2010. [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. [12] J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations, Ann.Inst. Henri Poincaré., 5 (1988), 365-405. [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. [14] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420. [15] O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$, Adv. Differential Equations, 3 (1998), 337-360. [16] O. Goubet and M. Hussein, Global attractor for the Davey-Stewartson system on $R^2$, Commun. Pure Appl. Anal., 8 (2009), 1555-1575. [17] O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schröinger equations in $L^2(R)$, Nonlinear Anal., 71 (2009), 317-320. [18] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence. [19] R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems, J. Differential equations, 145 (1998), 1-16. [20] R. Illner, H. Lange, B. Toomir and P. Zweifel, On quasi-linear Schrödinger-Poisson systems, Math. Meth. Appl. Sci., 20 (1997), 1223-1238. [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. [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. [23] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990. [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. [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. [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. [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. [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. [29] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997. [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.

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##### 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. [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. [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. [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. [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. [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [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. [8] T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Ellipses, Paris, 1990. [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. [10] A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson, Thèse, Université de Picardie Jules Verne, 2010. [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. [12] J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations, Ann.Inst. Henri Poincaré., 5 (1988), 365-405. [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. [14] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420. [15] O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$, Adv. Differential Equations, 3 (1998), 337-360. [16] O. Goubet and M. Hussein, Global attractor for the Davey-Stewartson system on $R^2$, Commun. Pure Appl. Anal., 8 (2009), 1555-1575. [17] O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schröinger equations in $L^2(R)$, Nonlinear Anal., 71 (2009), 317-320. [18] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence. [19] R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems, J. Differential equations, 145 (1998), 1-16. [20] R. Illner, H. Lange, B. Toomir and P. Zweifel, On quasi-linear Schrödinger-Poisson systems, Math. Meth. Appl. Sci., 20 (1997), 1223-1238. [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. [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. [23] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990. [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. [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. [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. [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. [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. [29] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997. [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.
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