Nonexistence of minimal blow-up solutions of equations iu t =-Δu-k(x)|u| 4/N u in N
Annales de l'I.H.P. Physique théorique, Tome 64 (1996) no. 1, pp. 33-85.
@article{AIHPA_1996__64_1_33_0,
     author = {Merle, Franck},
     title = {Nonexistence of minimal blow-up solutions of equations $iu_t = - \Delta u-k(x)|u|^{4/N} u$ in $\mathbb {R}^N$},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     pages = {33--85},
     publisher = {Gauthier-Villars},
     volume = {64},
     number = {1},
     year = {1996},
     zbl = {0846.35129},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPA_1996__64_1_33_0/}
}
TY  - JOUR
AU  - Merle, Franck
TI  - Nonexistence of minimal blow-up solutions of equations $iu_t = - \Delta u-k(x)|u|^{4/N} u$ in $\mathbb {R}^N$
JO  - Annales de l'I.H.P. Physique théorique
PY  - 1996
SP  - 33
EP  - 85
VL  - 64
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/item/AIHPA_1996__64_1_33_0/
LA  - en
ID  - AIHPA_1996__64_1_33_0
ER  - 
%0 Journal Article
%A Merle, Franck
%T Nonexistence of minimal blow-up solutions of equations $iu_t = - \Delta u-k(x)|u|^{4/N} u$ in $\mathbb {R}^N$
%J Annales de l'I.H.P. Physique théorique
%D 1996
%P 33-85
%V 64
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/item/AIHPA_1996__64_1_33_0/
%G en
%F AIHPA_1996__64_1_33_0
Merle, Franck. Nonexistence of minimal blow-up solutions of equations $iu_t = - \Delta u-k(x)|u|^{4/N} u$ in $\mathbb {R}^N$. Annales de l'I.H.P. Physique théorique, Tome 64 (1996) no. 1, pp. 33-85. http://archive.numdam.org/item/AIHPA_1996__64_1_33_0/

[1] H. Berestycki and P.L. Lions, Non linear scalar field equations I. Existence of a ground state; II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., Vol. 82, 1983, pp. 313-375. | Zbl

[1'] T. Cazenave and F. Weissler, The Cauchy problem in HS for nonlinear Schrödinger equation, preprint.

[2] J. Ginibre and G. Velo, On a class of nonlinear Schrödingre equations I, II. The Cauchy problem, general case, J. Func. Anal., Vol. 32, 1979, pp. 1-71. | MR | Zbl

[3] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two, Part I, Comm. Math. Phys., to appear. | MR | Zbl

[4] L. Glangetas and F. Merle, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two, Part II, Comm. Math. Phys., to appear. | MR | Zbl

[5] R.T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., Vol. 18, 1977, pp. 1794-1797. | MR | Zbl

[6] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Physique Théorique, Vol. 49, 1987, pp. 113-129. | Numdam | MR | Zbl

[7] M.K. Kwong, Uniqueness of positive solution of Δu - u + up = 0 in RN, Arch. Rational Mech. Anal., Vol. 105, 1989, pp. 243-266. | MR | Zbl

[8] M. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem and X.P. Wang, Stability of isotropic self-similar dynamics for scalar collapse, Phys. Rev. A, Vol. 46, 1992, pp. 4869-7876.

[9] F. Merle, Limit behavior of satured approximations of nonlinear Schrödinger equation, Comm. Math. Phys., Vol. 149, 1992, pp. 377-414. | MR | Zbl

[9'] F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., Vol. 129, 1990, pp. 223-240. | MR | Zbl

[10] F. Merle, Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power, Duke Math. J., Vol. 69, 1993, pp. 427-454. | MR | Zbl

[11] F. Merle, On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math., Vol. 45, 1992, pp. 203-254. | MR | Zbl

[12] F. Merle and Y. Tsutsumi, L2-concentration of blow-up solutions for the non-linear Schrödinger equation with the critical power nonlinearity, J. Diff. Eq., Vol. 84, 1990, pp. 205-214. | MR | Zbl

[13] T. Ozawa and Y. Tsutsumi, Blow-up for H1 solution for the nonlinear Schrödinger equation, preprint

[14] G.C. Papanicolaou, C. Sulem, P.L. Sulem and X.P. Wang, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B3, 1991, pp. 969-980. | MR

[15] V.V. Sobolev, V.S. Synach and V.E. Zakharov, Character of the singularity and stochastic phenomena in self-focusing, Zh. Eksp. Theor. Fiz, Pis'ma Red, Vol. 14, 1971, pp. 390-393.

[16] W.A. Strauss, Existence of solitary waves in hiher dimensions, Comm. Math. Phys., Vol. 55, 1977, p. 149-162. | MR | Zbl

[17] M.I. Weinstein, The nonlinear Schrödinger equation singularity formation stability and dispersion, AMS-SIAM Conference on the Connection between Infinite Dimensional and Finite Dimensional Dynamical Systems, July 1987. | Zbl

[18] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., Vol. 87, 1983, pp. 567-576. | MR | Zbl

[19] M.I. Weinstein, On the structure and formation of singularities in solutions to the nonlinear dispersive evolution equations, Comm. Partial Diff. Eq., Vol. 11, 1986, pp. 545-565. | MR | Zbl