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

Previous Article
Favard theory and fredholm alternative for disconjugate recurrent second order equations
CPAA Home
This Issue
Next Article
Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space

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

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]	Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121
[2]	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
[3]	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
[4]	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 - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[5]	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
[6]	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
[7]	Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[14]	S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593
[15]	Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[16]	Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019179
[17]	Mostafa Abounouh, Olivier Goubet. Regularity of the attractor for kp1-Burgers equation: the periodic case. Communications on Pure & Applied Analysis, 2004, 3 (2) : 237-252. doi: 10.3934/cpaa.2004.3.237
[18]	Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651
[19]	Daiwen Huang, Jingjun Zhang. Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1753-1773. doi: 10.3934/dcdss.2016073
[20]	Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028

American Institute of Mathematical Sciences