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

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 $H^{\frac{3}{2}-\epsilon}(-1, 1)$.

Citation: Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure and 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.

[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.

[3]

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

[4]

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

[5]

I. MoiseR. 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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[5]

I. MoiseR. 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.

[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.

[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.

[1]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete and 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 and 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 and 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 and 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 and 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 and Control Theory, 2022, 11 (2) : 559-581. 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (186)
  • HTML views (48)
  • Cited by (1)

Other articles
by authors

[Back to Top]