American Institute of Mathematical Sciences

Advanced
July  2017, 16(4): 1233-1252. doi: 10.3934/cpaa.2017060

Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect

Wided Kechiche

Unité de Recherche Multifractales et Ondelettes, Faculté des Sciences de Monastir, Université de Monastir, Av. de l'environnement, 5000 Monastir, Tunisie

Received  June 2016 Revised  February 2017 Published  April 2017

We consider a nonlinear Schrödinger equation with a delta-function impurity at the origin of the space domain. We study the asymptotic behavior of the solutions with the theory of infinite dynamical system. We first prove the existence of a global attractor in $H^1_0(-1, 1)$. We also establish that this global attractor is a compact subset of $H^{\frac{3}{2}-\epsilon}(-1, 1)$.

Keywords: Global attractor, nonlinear Schrödinger equation, regularity of the gobal attractor, dynamical systems, J. Ball's argument.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
References:
[1]

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

[2]

T. Cazenave, Semilinear Schrödinger Equations, vol 10, Courant Lectures Notes in Mathematics, NYU Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[3]

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

[4]

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

[5]

I. Moise, R. Rosa and X. Wang, Attractors for a non-compact semi-groups via energy equations, Nonlinearity, 11 (1998), 1369-1393.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar

[6]

C. Sulem and P. L Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.  Google Scholar

[7]

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

show all references

References:
[1]

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

[2]

T. Cazenave, Semilinear Schrödinger Equations, vol 10, Courant Lectures Notes in Mathematics, NYU Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[3]

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

[4]

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

[5]

I. Moise, R. Rosa and X. Wang, Attractors for a non-compact semi-groups via energy equations, Nonlinearity, 11 (1998), 1369-1393.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar

[6]

C. Sulem and P. L Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.  Google Scholar

[7]

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

[1]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031
[2]

Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121
[3]

Olivier Goubet, Ezzeddine Zahrouni. Global attractor for damped forced nonlinear logarithmic Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2933-2946. doi: 10.3934/dcdss.2020393
[4]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[5]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[6]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013
[7]

Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206
[8]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[9]

Salah Missaoui. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system. Communications on Pure & Applied Analysis, 2022, 21 (2) : 567-584. doi: 10.3934/cpaa.2021189
[10]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[11]

Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695
[12]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[13]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[14]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[15]

Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695
[16]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305
[17]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[18]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
[19]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[20]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (165)
  • HTML views (48)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy