# 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 & 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 $R$,, \emph{Appl. Math. Lett.}, 12 (1999), 45. doi: 10.1016/S0893-9659(98)00170-0. Google Scholar [2] J. Ball, Global attractors for damped semilinear wave equations,, \emph{Partial Differential Equations and Applications, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. Google Scholar [3] K. Benmlih, Stationary solutions for schrödinger-Poisson system in $\mathbbR^3$,, \emph{Proceedings of the 2002 Fez Conference on Partial Differential Equations}, (2002), 65. Google Scholar [4] K. Benmlih and O. Kavian, Existence and asymptotic behaviour of standing waves for quasilinear Schröinger-Poisson systems in $R^3$,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'aire}, 25 (2008), 449. Google Scholar [5] B. Bongioanni et J. L. Torrea, Sobolev spaces associated to the harmonic oscillator,, \emph{Proc. Indian. Acad. Sci. (Math. Sci.)}, 116 (2006), 337. doi: 10.1007/BF02829750. Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar [7] T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar [8] T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires,, Ellipses, (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,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 121. Google Scholar [10] A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson,, Th\ese, (2010). Google Scholar [11] A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson sur un domaine borné de $\mathbbR^3$,, \emph{Ann. Math. Blaise Pascal}, 17 (2010), 199. Google Scholar [12] J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations,, \emph{Ann.Inst. Henri Poincar\'e.}, 5 (1988), 365. 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,, \emph{RAIRO Mod\'e. Math. Anal. Num\'e}, 23 (1989), 433. Google Scholar [14] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99. doi: 10.1080/00036819608840420. Google Scholar [15] O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$,, \emph{Adv. Differential Equations}, 3 (1998), 337. Google Scholar [16] O. Goubet and M. Hussein, Global attractor for the Davey-Stewartson system on $R^2$,, \emph{Commun. Pure Appl. Anal.}, 8 (2009), 1555. Google Scholar [17] O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schröinger equations in $L^2(R)$,, \emph{Nonlinear Anal.}, 71 (2009), 317. Google Scholar [18] J. Hale, Asymptotic Behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988). Google Scholar [19] R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems,, \emph{J. Differential equations}, 145 (1998), 1. Google Scholar [20] R. Illner, H. Lange, B. Toomir and P. Zweifel, On quasi-linear Schrödinger-Poisson systems,, \emph{Math. Meth. Appl. Sci.}, 20 (1997), 1223. Google Scholar [21] M. Jolly, T. Sadigov and E. Titi, A determining form for the damped driven nonlinear Schröinger equationfourier modes case,, \emph{J. Differential Equations}, 258 (2015), 2711. Google Scholar [22] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 2 (1995), 357. doi: 10.1007/BF01261181. Google Scholar [23] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer, (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, (1030). 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 $L^2(T)$,, \emph{Dyn. Partial Differ. Equ.}, 6 (2009). Google Scholar [26] F. Nier, Schrödinger-Poisson systems in dimension $d\leq3$, the whole space case,, \emph{Proceedings of the Royal Society of Edinburgh}, 123A (1993), 1179. 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,, \emph{Indiana Univ. Math. J.}, 47 (1998). Google Scholar [28] G. Raugel, Global attractors in partial differential equations,, in \emph{Handbook of Dynamical Systems}, (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. Google Scholar [29] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). Google Scholar [30] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors,, \emph{Physica D}, 88 (1995), 167. doi: 10.1016/0167-2789(95)00196-B. Google Scholar

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##### References:
 [1] N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$,, \emph{Appl. Math. Lett.}, 12 (1999), 45. doi: 10.1016/S0893-9659(98)00170-0. Google Scholar [2] J. Ball, Global attractors for damped semilinear wave equations,, \emph{Partial Differential Equations and Applications, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. Google Scholar [3] K. Benmlih, Stationary solutions for schrödinger-Poisson system in $\mathbbR^3$,, \emph{Proceedings of the 2002 Fez Conference on Partial Differential Equations}, (2002), 65. Google Scholar [4] K. Benmlih and O. Kavian, Existence and asymptotic behaviour of standing waves for quasilinear Schröinger-Poisson systems in $R^3$,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'aire}, 25 (2008), 449. Google Scholar [5] B. Bongioanni et J. L. Torrea, Sobolev spaces associated to the harmonic oscillator,, \emph{Proc. Indian. Acad. Sci. (Math. Sci.)}, 116 (2006), 337. doi: 10.1007/BF02829750. Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar [7] T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar [8] T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires,, Ellipses, (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,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 121. Google Scholar [10] A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson,, Th\ese, (2010). Google Scholar [11] A. Dabaa, Comportement asymptotique des solutions d'un système d'équations de Schrödinger-Poisson sur un domaine borné de $\mathbbR^3$,, \emph{Ann. Math. Blaise Pascal}, 17 (2010), 199. Google Scholar [12] J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations,, \emph{Ann.Inst. Henri Poincar\'e.}, 5 (1988), 365. 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,, \emph{RAIRO Mod\'e. Math. Anal. Num\'e}, 23 (1989), 433. Google Scholar [14] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99. doi: 10.1080/00036819608840420. Google Scholar [15] O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$,, \emph{Adv. Differential Equations}, 3 (1998), 337. Google Scholar [16] O. Goubet and M. Hussein, Global attractor for the Davey-Stewartson system on $R^2$,, \emph{Commun. Pure Appl. Anal.}, 8 (2009), 1555. Google Scholar [17] O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schröinger equations in $L^2(R)$,, \emph{Nonlinear Anal.}, 71 (2009), 317. Google Scholar [18] J. Hale, Asymptotic Behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988). Google Scholar [19] R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems,, \emph{J. Differential equations}, 145 (1998), 1. Google Scholar [20] R. Illner, H. Lange, B. Toomir and P. Zweifel, On quasi-linear Schrödinger-Poisson systems,, \emph{Math. Meth. Appl. Sci.}, 20 (1997), 1223. Google Scholar [21] M. Jolly, T. Sadigov and E. Titi, A determining form for the damped driven nonlinear Schröinger equationfourier modes case,, \emph{J. Differential Equations}, 258 (2015), 2711. Google Scholar [22] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 2 (1995), 357. doi: 10.1007/BF01261181. Google Scholar [23] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer, (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, (1030). 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 $L^2(T)$,, \emph{Dyn. Partial Differ. Equ.}, 6 (2009). Google Scholar [26] F. Nier, Schrödinger-Poisson systems in dimension $d\leq3$, the whole space case,, \emph{Proceedings of the Royal Society of Edinburgh}, 123A (1993), 1179. 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,, \emph{Indiana Univ. Math. J.}, 47 (1998). Google Scholar [28] G. Raugel, Global attractors in partial differential equations,, in \emph{Handbook of Dynamical Systems}, (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. Google Scholar [29] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). Google Scholar [30] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors,, \emph{Physica D}, 88 (1995), 167. doi: 10.1016/0167-2789(95)00196-B. Google Scholar
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