Equation de Schrödinger en milieu inhomogène

Banica, Manuela Valeria

Directeur de thèse : Gérard, Patrick

Membres du jury : Cazenave, Thierry ; Gérard, Patrick ; Métivier, Guy ; Planchon, Fabrice ; Saut, Jean-Claude ; Zworski, Maciej

Thèses d'Orsay, no. 633 (2003) , 106 p.

Export
Comment citer

@phdthesis{BJHTUP11_2003__0633__P0_0,
     author = {Banica, Manuela Valeria},
     title = {Equation de {Schr\"odinger} en milieu inhomog\`ene},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud U.F.R. Scientifique d'Orsay},
     number = {633},
     year = {2003},
     language = {fr},
     url = {http://www.numdam.org/item/BJHTUP11_2003__0633__P0_0/}
}

TY  - BOOK
AU  - Banica, Manuela Valeria
TI  - Equation de Schrödinger en milieu inhomogène
T3  - Thèses d'Orsay
PY  - 2003
IS  - 633
PB  - Université de Paris-Sud U.F.R. Scientifique d'Orsay
UR  - http://www.numdam.org/item/BJHTUP11_2003__0633__P0_0/
LA  - fr
ID  - BJHTUP11_2003__0633__P0_0
ER  -

%0 Book
%A Banica, Manuela Valeria
%T Equation de Schrödinger en milieu inhomogène
%S Thèses d'Orsay
%D 2003
%N 633
%I Université de Paris-Sud U.F.R. Scientifique d'Orsay
%U http://www.numdam.org/item/BJHTUP11_2003__0633__P0_0/
%G fr
%F BJHTUP11_2003__0633__P0_0

Banica, Manuela Valeria. Equation de Schrödinger en milieu inhomogène. Thèses d'Orsay, no. 633 (2003), 106 p. http://numdam.org/item/BJHTUP11_2003__0633__P0_0/

Bibliographie
Cité par

[1] C. Antonini, Lower bounds for the $𝕃^{2}$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Diff. Integral Eq. 15 (2002), no. 6, 749-768. | MR | Zbl

[2] M. Avellaneda, C. Bardos and J. Rauch, Contrôlabilité exacte, homogénéisation et localisation d'ondes dans un milieu non-homogène, Asymptotic Analysis 5 (1992), no. 6, 481-494. | MR | Zbl

[3] H. Bahouri, J. Y. Chemin, Equations d'ondes quasi-linéaires et estimations de Strichartz, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 9, 803-806. | MR | Zbl

[4] V. Banica, Contre-exemples aux inégalités de Strichartz pour des équations d'évolution à coefficients singuliers, mémoire de DEA, Université de Paris Sud (1999).

[5] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I.Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107-156. | MR | Zbl | DOI

[6] H. Brézis, T. Gallouët, Nonlinear Schrödinger evolution equation, Nonlinear Analysis, Theory Methods Appl. 4 (1980), no. 4, 677-681. | MR | Zbl | DOI

[7] N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, to appear in Amer. J. Math.. | MR | Zbl

[8] N. Burq, P. Gérard, N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^{d}$ , Math. Res. Lett. 9 (2002), no. 2-3, 323-335. | MR | Zbl | DOI

[9] N. Burq, P. Gérard, N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal. 13 (2003), 1-19. | MR | Zbl | DOI

[10] N. Burq, P. Gérard, N. Tzvetkov, Inégalités de Sogge bilinéaires et équation de Schrödinger non-linéaire, Séminaire Equations aux Dérivées partielles, Ecole Polytechnique, Palaiseau, mars 2003. | MR | Numdam | Zbl

[11] C. Castro, E. Zuazua, Concentration and lack of observability of waves in highly heterogeneous media, Arch. Ration. Mech. Anal. 164 (2002), no. 1, 39-72. | MR | Zbl | DOI

[12] T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos 26, Instituto de Matemática-UFRJ, Rio de Janeiro, RJ (1996).

[13] T. Cazenave, Equation de Schrödinger non-linéaire en dimension deux, Proc. Roy. Soc. Edinburg, 84A, 327-346. | MR | Zbl | DOI

[14] T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $ℍ^{1}$ , Manuscripta Math. 61 (1988), no. 4, 477-494. | MR | Zbl | DOI

[15] M. Christ, J. Colliander, T. Tao, Asymptotic, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, to appear in Amer. J. Math.. | MR | Zbl

[16] W. Craig, T. Kappeler, W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), no. 8, 769-860. | MR | Zbl | DOI

[17] S. Doi, Remarks on the Cauchy problem for Schrödinger-type equations, Comm. Partial Differential Equations 21 (1996), no. 1-2, 163-178. | MR | Zbl

[18] J. Ginibre, Introduction aux equations de Schrödinger non linéaires, Editions Paris XI (1994).

[19] J. Ginibre, G. Velo, On a class of Schrödinger equations. I. The Cauchy problem, a general case, J. Funct. Anal. 32 (1979), no. 1, 1-32. | MR | Zbl

[20] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, Ann. I. H. P. Analyse non-linéaire 2 (1985), no. 4, 309-327. | MR | Zbl | Numdam

[21] J. Ginibre, G. Velo, Generalized Strichartz Inequalities for the Wave Equation, J. Funct. Analysis 133 (1995), no. 1, 50-68. | MR | Zbl | DOI

[22] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794-1797. | MR | Zbl | DOI

[23] L. Kapitanski, Some generalisations of the Strichartz-Brenner inequality, Leningrad Math. J. 1 (1990), 693-726. | Zbl | MR

[24] L. Kapitanski, Y. Safarov, A Parametrix for the Nonstationary Schrödinger Equation, Differential Operators and Spectral Theory ; Amer. Math. Soc. Transl. Ser. 2, 189, (1999), 139-148. | MR | Zbl

[25] L. Kapitanski, Y. Safarov, Dispersive smoothing for Schrödinger equations, Math. Res. Lett. 3 (1996), no. 1, 77-91. | MR | Zbl | DOI

[26] T. Kato, On nonlinear Schrödinger equations, Ann. I. H. P. Physique Théorique 46 (1987), no. 1, 113-129. | MR | Zbl | Numdam

[27] O. Kavian, A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 299 (1987), no. 1, 193-203. | MR | Zbl

[28] M. Keel, T. Tao, Endpoint Strichartz Estimates, Amer. J. Math. 120 (1998), no. 5, 955-980. | MR | Zbl | DOI

[29] C. Kenig, G. Ponce, L. Vega, Smoothing Effects and Local Existence Theory for the Generalized Nonlinear Schrödinger Equations, Invent. Math. 134 (1998), no. 3, 489-545. | MR | Zbl | DOI

[30] C. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617-633. | MR | Zbl | DOI

[31] S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. | MR | Zbl | DOI

[32] H. Koch, N. Tzvetkov, On the local well-posedness on the Benjamin-Ono equation, to appear in I. M. R. N.. | MR | Zbl

[33] M. K. Kwong, Uniqueness of positive solutions of $Δ u - u + u^{p} = 0$ in $ℝ^{N}$ , Arch. Rat. Mech. Ann. 105 (1989), no. 3, 243-266. | MR | Zbl

[34] G. Lebeau, Non linear optic and supercritical wave equation. Hommage à Pascal Laubin, Bull. Soc. Roy. Sci. Liège 70 (2001), no. 4-6, 267-306 (2002). | MR | Zbl

[35] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145. | MR | Zbl | Numdam | DOI

[36] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincare Anal. Non Linéaire 1 (1984), no. 4, 223-283. | MR | Zbl | Numdam | DOI

[37] M. Maris, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures. Appl. 81 (2002), 1207-1239. | MR | Zbl | DOI

[38] F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), no. 2, 427-454. | MR | Zbl | DOI

[39] F. Merle, P. Raphaël, Blow-up dynamic and upper bound on blow-up rate for critical non linear Schrödinger equation, Université de Cergy-Pontoise, prépublication (2003). | MR | Zbl

[40] F. Merle, P. Raphaël, On blow-up profile for critical non linear Schrödinger equation, Université de Cergy-Pontoise, prépublication (2003).

[41] G. Métivier, Exemples d'instabilités pour des équations d'ondes non linéaires [d'après G. Lebeau], Séminaire Bourbaki, 55ème année, 2002-2003, no. 911. | MR | Zbl | Numdam

[42] S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations, Duke Math. J. 91 (1998), no. 2, 393-408. | MR | Zbl | DOI

[43] T. Ogawa, T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger equations, J. Math. Anal. Appl. 155 (1991), no. 2, 531-540. | MR | Zbl | DOI

[44] T. Ogawa, Y. Tsutsumi, Blow-up solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary conditions, Springer Lecture Notes in Math. 1450 (1990), 236-251. | MR | Zbl

[45] M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV : Analysis of Operators, Academic Press, New York (1978). | MR

[46] H.F. Smith, A parametrix construction for wave equation with $C^{1, 1}$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797-835. | MR | Zbl | Numdam | DOI

[47] H.F. Smith, C.D. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett. 1 (1994), no. 6, 729-737. | MR | Zbl | DOI

[48] H.F. Smith, C.D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 (1995), no. 4, 879-916. | MR | Zbl | DOI

[49] C. Sogge, Fourier integrals in classical analysis, Cambridge tracts in mathematics (1993). | MR | Zbl

[50] C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), no. 1, 43-65. | MR | Zbl | DOI

[51] G. Staffilani, D. Tataru, Strichartz Estimates for a Schrödinger Operator with nonsmooth Coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337-1372. | MR | Zbl | DOI

[52] R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705-714. | MR | Zbl | DOI

[53] C. Sulem, P. L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Math. Sciences, 139, Springer-Verlag, New York (1992). | MR | Zbl

[54] D. Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), no. 2, 349-376. | MR | Zbl | DOI

[55] P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 177-178. | MR | Zbl | DOI

[56] M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR 275 (1984), no. 4, 780-783. | MR | Zbl

[57] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolate estimates, Comm. Math. Phys. 87 (1983), no. 4, 567-576. | MR | Zbl | DOI

[58] M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Part. Diff. Eq. 11 (1986), no. 5, 545-565. | MR | Zbl | DOI

[59] M. I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, Siam. J. Math. Anal. 16 (1985), no. 3, 472-491. | MR | Zbl | DOI

[60] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415-426. | MR | Zbl | DOI

[61] V. E. Zakharov, Collapse of Lagmuir waves, Sov. Phys. JETP 35 (1972), 908-914.

[1] M. Avellaneda, C. Bardos, J. Rauch, Contrôlabilité exacte, homogénéisation et localisation d'ondes dans un milieu non-homogène, Asymptot. Anal. 5 (1992), no. 6, 481-494. | MR | Zbl

[2] N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., to appear. | MR | Zbl

[3] R. H. Cameron, Analytic functions of absolutely convergent generalized trigonometric sums, Duke Math. J. 3 (1937), 682-688. | MR | Zbl | DOI

[4] C. Castro, E. Zuazua, Concentration and lack of observability of waves in highly heterogeneous media, Arch. Ration. Mech. Anal. 164 (2002), no. 1, 30-72. | MR | Zbl | DOI

[5] W. Craig, T. Kappeler, W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), no. 8, 769-860. | MR | Zbl | DOI

[6] S. Doi, Remarks on the Cauchy problem for Schrödinger-type equations, Comm. Partial Differential Equations 21 (1996), no. 1-2, 163-178. | MR | Zbl

[7] I.M. Gelfand, D.A. Raikov and G.E. Chilov, Les anneaux normés commutatifs, Monographies internationales de mathématiques modernes, Gauthier-Villars Paris (1964). | Zbl

[8] J. Ginibre, G. Velo, Generalized Strichartz Inequalities for the Wave Equation, J. Funct. Anal. 133 (1995), no. 1, 50-68. | MR | Zbl | DOI

[9] L. Kapitanski, Y. Safarov, Dispersive smoothing for Schrödinger equations, Math. Res. Lett. 3 (1996), no. 1, 77-91. | MR | Zbl | DOI

[10] C. E. Kenig, G. Ponce, L. Vega, Smoothing Effects and Local Existence Theory for the Generalized Nonlinear Schrödinger Equations, Invent. Math. 134 (1998), no. 3, 489-545. | MR | Zbl | DOI

[11] W. Magnus, S. Winkler, Hill's equation, Interscience tracts in pure and applies mathematics (1966). | MR | Zbl

[12] H. R. Pitt, A theorem on absolutely convergent trigonometric series, J. Math. Phys. 16 (1938), 191-195. | Zbl | JFM | DOI

[13] G. Staffilani and D. Tataru, Strichartz Estimates for a Schrödinger Operator with nonsmooth Coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337-1372. | MR | Zbl | DOI

[14] R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705-714. | MR | Zbl | DOI

[15] P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 177-178. | MR | Zbl | DOI

[1] J. Bourgain, Fourier transformation restriction phenomena for certain lattice subsets and application to the nonlinear evolution equations I. Schrödinger equations, Geom. and Funct. Anal. 3 (1993), no. 2, 107-156. | MR | Zbl | DOI

[2] N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, to appear in Amer. J. Math.. | MR | Zbl

[3] N. Burq, P. Gérard, N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $𝕊^{d}$ , Math. Res. Lett. 9 (2002), no. 2-3, 323-335. | MR | Zbl | DOI

[4] N. Burq, P. Gérard, N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal. 13 (2003), 1-19. | MR | Zbl | DOI

[5] N. Burq, P. Gérard, N. Tzvetkov, Inégalités de Sogge bilinéaires et équation de Schrödinger non-linéaire, Séminaire Equations aux Dérivées partielles, École Polytechnique, Palaiseau, mars 2003. | MR | Zbl

[6] T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $ℍ^{s}$ , Nonlinear Analysis 14 (1990), no. 10, 807-836. | MR | Zbl | DOI

[7] M. Christ, J. Colliander, T. Tao, Asymptotic, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, to appear in Amer. J. Math.. | MR | Zbl

[8] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, Ann. I. H. P. Ann. non-lin. 2 (1985), no. 4, 309-327. | MR | Zbl | Numdam

[9] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge University Press (1955). | MR | JFM

[10] T. Kato, On nonlinear Schrödinger equations, Ann. I. H. P. Phys. Th. 46 (1987), no. 1, 113-129. | MR | Zbl | Numdam

[11] C. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617-633. | MR | Zbl | DOI

[12] Koch, N. Tzvetkov, On the local well-posedness on the Benjamin-Ono equation, to appear in I. M. R. N.. | MR | Zbl

[13] G. Lebeau, Optique non linéaire et ondes sur critiques, Séminaire Equations aux Dérivées partielles, 1999-2000, Exp. No. IV, 13 pp., Sémin. Équ. Dériv. Partielles, Ecole Polytechnique, Palaiseau (2000). | MR | Zbl

[14] G. Métivier, Exemples d'instabilités pour des équations d'ondes non linéaires [d'après G. Lebeau], Séminaire Bourbaki, 55ème année, 2002-2003, no. 911. | MR | Zbl | Numdam

[15] C. Sogge, Fourier integrals in classical analysis, Cambridge tracts in mathematics (1993). | MR | Zbl

[16] C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), no. 1, 43-65. | MR | Zbl | DOI

[17] C. Sulem, P. L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Math. Sciences, 139, Springer-Verlag, New York (1992). | MR | Zbl

[18] R. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705-714. | MR | Zbl | DOI

[19] K. Yajima, Existence of solutions for Schrödinger evolution equation, Comm. Math. Phys. 110 (1987), no. 3, 415-426. | MR | Zbl | DOI

[1] C. Antonini, Lower bounds for the $𝕃^{2}$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Diff. Integral Eq. 15 (2002), no. 6, 749-768. | MR | Zbl

[2] H. Brézis, T. Gallouët, Nonlinear Schrödinger evolution equation, Nonlinear Analysis, Theory Methods Appl. 4 (1980), no. 4, 677-681. | MR | Zbl | DOI

[3] N. Burq, P. Gerard, N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal. 13 (2003), 1-19. | MR | Zbl | DOI

[4] T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos 26, Instituto de Matemática-UFRJ, Rio de Janeiro, RJ (1996).

[5] I. Gallagher, P. Gérard, Profile decomposition for the wave equation outside a convex obstacle, J. Math. Pures Appl. (9) 80 (2001), no. 1, 1-49. | MR | Zbl | DOI

[6] J. Ginibre, G. Velo, On a class of Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1-71. | MR | Zbl

[7] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794-1797. | MR | Zbl | DOI

[8] T. Kato, On nonlinear Schrödinger equations, Ann. I. H. P. Physique Théorique 46 (1987), no. 1, 113-129. | MR | Zbl | Numdam

[9] O. Kavian, A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 299 (1987), no. 1, 193-203. | MR | Zbl

[10] M. K. Kwong, Uniqueness of positive solutions of $Δ u - u + u^{p} = 0$ in $ℝ^{N}$ , Arch. Rat. Mech. Ann. 105 (1989), no. 3, 243-266. | MR | Zbl

[11] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145. | MR | Zbl | Numdam | DOI

[12] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. | MR | Zbl | Numdam | DOI

[13] M. Marcus, V. J. Mizel, Nemytskij operators in Sobolev spaces, Arch. Rat. Mech. Anal. 51 (1973), 347-370. | MR | Zbl | DOI

[14] M. Maris, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures. Appl. 81 (2002), 1207-1239. | MR | Zbl | DOI

[15] F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), no. 2, 427-454. | MR | Zbl | DOI

[16] F. Merle, Asymptotics for $L^{2}$ minimal blow-up solutions of critical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 5, 553-565 | MR | Zbl | DOI | Numdam

[17] F. Merle, P. Raphaël, Blow-up dynamic and upper bound on blow-up rate for critical non linear Schrödinger equation, Université de Cergy-Pontoise, preprint (2003). | MR | Zbl

[18] F. Merle, P. Raphaël, On blow-up profile for critical non linear Schrödinger equation, Université de Cergy-Pontoise, preprint (2003).

[19] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 3 (1959), no. 13, 115-162. | MR | Zbl | Numdam

[20] T. Ogawa, T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger equations, J. Math. Anal. Appl. 155 (1991), no. 2, 531-540. | MR | Zbl | DOI

[21] T. Ogawa, Y. Tsutsumi, Blow-up solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary conditions, Springer Lecture Notes in Math. 1450 (1990), 236-251. | MR | Zbl

[22] M. Reed, B. Simon, Methods of modern mathematical Physics IV : Analysis of Operators, Academic Press, New York (1978). | MR

[23] C. Sulem, P. L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Math. Sciences, 139, Springer-Verlag, New York (1992). | MR | Zbl

[24] M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR 275 (1984), no. 4, 780-783. | MR | Zbl

[25] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolate estimates, Comm. Math. Phys. 87 (1983), no. 4, 567-576. | MR | Zbl | DOI

[26] M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Part. Diff. Eq. 11 (1986), no. 5, 545-565. | MR | Zbl | DOI

[27] M. I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, Siam. J. Math. Anal. 16 (1985), no. 3, 472-491. | MR | Zbl | DOI

[28] V. E. Zakharov, Collapse of Lagmuir waves, Sov. Phys. JETP 35 (1972), 908-914.