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

Banica, Valeria

Journées équations aux dérivées partielles (2003), article no. 1, 14 p.

We concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound from below the blow-up rate for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than ${(T-t)}^{-1}$, the expected one. Moreover, we state that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.

[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

[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. | MR | Zbl

[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

[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

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

[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 $\Delta u-u+u^p=0$ 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. Poincarés Anal. Non Linéaire 1 (1984), no. 2, 109-145. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

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

[14] 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

[15] 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).

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

[17] 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

[18] 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

[19] 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

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

[21] 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

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

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