Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Volumn 3, Issue 1, 2004, Pages 139-170

  Banica, Valeria ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 28444487069     PISSN: 0391173X     EISSN: 20362145     Source Type: Journal    
DOI: None     Document Type: Article

References (30)

1. 2 minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Differential Integral Equations 15 (2002), 749-768.
2. H. Brézis - T. Gallouët, Nonlinear Schrödinger evolution equation, Nonlinear Anal. 4 (1980), 677-681.
3. 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.
4. T. Cazenave, “An introduction to nonlinear Schrödinger equations”, Textos de Métodos Matemáticos 26, Instituto de Matemática-UFRJ, Rio de Janeiro, 1996.
5. 1, Manuscripta Math. 61 (1988), 477-494.
6. T. Cazenave - F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, In: “Nonlinear semigroups, partial differential equations and attractors” (Washington, 1987), Lecture Notes in Math., 1394, Springer, Berlin, 1989, pp 18-29.
7. I. Gallagher - P. Gérard, Profile decomposition for the wave equation outside a convex obstacle, J. Math. Pures Appl. 80 (2001), 1-49.
8. J. Ginibre - G. Velo, On a class of Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1-71.
9. R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797.
10. T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré - Physique Théorique 46 (1987), 113-129.
11. O. Kavian, A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 299 (1987), 193-203.
12. N, Arch. Ration. Mech. Anal. 105 (1989), 243-266.
13. 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), 109-145.
14. 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), 223-283.
15. M. Marcus - V. J. Mizel, Nemytskij operators in Sobolev spaces, Arch. Ration. Mech. Anal. 51 (1973), 347-370.
16. M. Maris, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures. Appl. 81 (2002), 1207-1239.
17. F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), 427-454.
18. 2 minimal blow-up solutions of critical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré - Anal. Non Linéaire 13 (1996), 553-565.
19. F. Merle - P. Raphaël, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), 591-642.
20. F. Merle - P. Raphaël, On blow-up profile for critical non linear Schrödinger equation, Université de Cergy-Pontoise, preprint (2003).
21. L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) (1959), 115-162.
22. T. Ogawa - T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger equations, J. Math. Anal. Appl. 155 (1991), 531-540.
23. 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.
24. M. Reed - B. Simon, “Methods of modern mathematical Physics IV: Analysis of Operators”, Academic Press, New York, 1978.
25. C. Sulem - P. L. Sulem, “The nonlinear Schrödinger equation. Self-focusing and wave collapse”, Applied Math. Sciences, 139, Springer-Verlag, New York, 1992.
26. M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR 275 (1984), 780-783.
27. M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolate estimates,Comm. Math. Phys. 87 (1983), 567-576.
28. M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11 (1986), 545-565.
29. M. I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472-491.
30. V. E. Zakharov, Collapse of Lagmuir waves, Sov. Phys. JETP 35 (1972), 908-914.