On a nonlinear Schrödinger equation with a localizing effect

Pascal Bégout; Jesús Ildefonso Díaz

doi:10.1016/j.crma.2006.01.027

Partial Differential Equations

On a nonlinear Schrödinger equation with a localizing effect
[Sur une équation de Schrödinger non-linéaire avec un effet localisant]

Présenté par : Haïm Brezis

Pascal Bégout ¹ ; Jesús Ildefonso Díaz ²

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4, place Jussieu, 75252 Paris cedex 05, France
² Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, 28040 Madrid, Spain

Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 459-463.

Résumé
Abstract

Nous considérons l'équation de Schrödinger non-linéaire associée à un potentiel singulier de la forme $a {| u |}^{- (1 - m)} u + b u$ , avec $m \in (0, 1)$ , sur un domaine éventuellement non borné. Nous employons des méthodes d'énergie appropriées pour montrer que si $Re (a) + Im (a) > 0$ et si les données (initiale et source) ont un support compact alors toute solution doit également avoir un support compact pour tout $t > 0$ . Cette propriété contraste avec le comportement des solutions associées aux potentiels réguliers $(m ⩾ 1)$ . Des résultats similaires sont également établis pour le problème stationnaire associé et pour les solutions auto-similaires sur l'espace entier et le potentiel $a {| u |}^{- (1 - m)} u$ . L'existence des solutions est obtenue par des méthodes de compacité sous certaines conditions.

We consider the nonlinear Schrödinger equation associated to a singular potential of the form $a {| u |}^{- (1 - m)} u + b u$ , for some $m \in (0, 1)$ , on a possible unbounded domain. We use some suitable energy methods to prove that if $Re (a) + Im (a) > 0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t > 0$ . This property contrasts with the behavior of solutions associated to regular potentials $(m ⩾ 1)$ . Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a {| u |}^{- (1 - m)} u$ . The existence of solutions is obtained by some compactness methods under additional conditions.

Reçu le : 2006-01-20
Publié le : 2006-03-06

DOI : 10.1016/j.crma.2006.01.027

Affiliations des auteurs :

Pascal Bégout ¹ ; Jesús Ildefonso Díaz ²

@article{CRMATH_2006__342_7_459_0,
     author = {Pascal B\'egout and Jes\'us Ildefonso D{\'\i}az},
     title = {On a nonlinear {Schr\"odinger} equation with a localizing effect},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--463},
     publisher = {Elsevier},
     volume = {342},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.027},
     language = {en},
}

TY  - JOUR
AU  - Pascal Bégout
AU  - Jesús Ildefonso Díaz
TI  - On a nonlinear Schrödinger equation with a localizing effect
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 459
EP  - 463
VL  - 342
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2006.01.027
LA  - en
ID  - CRMATH_2006__342_7_459_0
ER  -

%0 Journal Article
%A Pascal Bégout
%A Jesús Ildefonso Díaz
%T On a nonlinear Schrödinger equation with a localizing effect
%J Comptes Rendus. Mathématique
%D 2006
%P 459-463
%V 342
%N 7
%I Elsevier
%R 10.1016/j.crma.2006.01.027
%G en
%F CRMATH_2006__342_7_459_0

Pascal Bégout; Jesús Ildefonso Díaz. On a nonlinear Schrödinger equation with a localizing effect. Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 459-463. doi : 10.1016/j.crma.2006.01.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.027/

Bibliographie
Cité par

[1] S.N. Antontsev; J.I. Díaz; H.B. de Oliveira Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem, J. Math. Fluid Mech., Volume 6 (2004), pp. 439-461

[2] S.N. Antontsev; J.I. Díaz; S. Shmarev Energy Methods for Free Boundary Problems, Birkhäuser Boston Inc., Boston, MA, 2002

[3] P. Bégout, J.I. Díaz, Localizing estimates of the support of solutions of some nonlinear Schrödinger equations, in press

[4] P. Bégout, J.I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, in press

[5] H. Brezis; T. Kato Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., Volume 58 (1979), pp. 137-151

[6] T. Cazenave Semilinear Schrödinger Equations, Courant Lecture Notes in Math., vol. 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003

[7] O. Kavian; F.B. Weissler Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., Volume 41 (1993), pp. 151-173

[8] B.J. LeMesurier Dissipation at singularities of the nonlinear Schrödinger equation through limits of regularizations, Physica D, Volume 138 (2000), pp. 334-343

[9] V. Liskevitch; P. Stollmann Schrödinger operators with singular complex potentials as generators: existence and stability, Semigroup Forum, Volume 60 (2000), pp. 337-343

[10] P. Rosenau; S. Schochet Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit, Chaos, Volume 15 (2005), pp. 1-18

[11] C. Sulem; P.-L. Sulem The Nonlinear Schrödinger Equation, Springer-Verlag, New York, 1999

[12] I.I. Vrabie Compactness Methods for Nonlinear Evolutions, Pitman Monogr. Surveys Pure Appl. Math., vol. 75, Longman Scientific & Technical, Harlow, 1987

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

On the confinement of a viscous fluid by means of a feedback external field

S.N. Antontsev; J.I. Dı́az; H.B. de Oliveira

C. R. Méca (2002)