The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Nonexistence of minimal blow-up solutions of equations i u t = - Δ u - k ( x ) | u | 4 / N u in N

Franck Merle

Annales de l'I.H.P. Physique théorique (1996)

  • Volume: 64, Issue: 1, page 33-85
  • ISSN: 0246-0211

How to cite

top

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 64.1 (1996): 33-85. <http://eudml.org/doc/76708>.

@article{Merle1996,
author = {Merle, Franck},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {nonlinear Schrödinger equation with critical exponent; concentration properties; blow-up time; black holes},
language = {eng},
number = {1},
pages = {33-85},
publisher = {Gauthier-Villars},
title = {Nonexistence of minimal blow-up solutions of equations $iu_t = - \Delta u-k(x)|u|^\{4/N\} u$ in $\mathbb \{R\}^N$},
url = {http://eudml.org/doc/76708},
volume = {64},
year = {1996},
}

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
PB - Gauthier-Villars
VL - 64
IS - 1
SP - 33
EP - 85
LA - eng
KW - nonlinear Schrödinger equation with critical exponent; concentration properties; blow-up time; black holes
UR - http://eudml.org/doc/76708
ER -

References

top
  1. [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. Zbl0533.35029
  2. [1'] T. Cazenave and F. Weissler, The Cauchy problem in HS for nonlinear Schrödinger equation, preprint. Zbl0696.35153
  3. [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. Zbl0396.35028MR533219
  4. [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. Zbl0808.35137MR1262194
  5. [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. Zbl0808.35138MR1262202
  6. [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. Zbl0372.35009MR460850
  7. [6] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Physique Théorique, Vol. 49, 1987, pp. 113-129. Zbl0632.35038MR877998
  8. [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. Zbl0676.35032MR969899
  9. [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. 
  10. [9] F. Merle, Limit behavior of satured approximations of nonlinear Schrödinger equation, Comm. Math. Phys., Vol. 149, 1992, pp. 377-414. Zbl0756.35094MR1186035
  11. [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. Zbl0707.35021MR1048692
  12. [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. Zbl0808.35141MR1203233
  13. [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. Zbl0767.35084MR1139066
  14. [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. Zbl0722.35047MR1047566
  15. [13] T. Ozawa and Y. Tsutsumi, Blow-up for H1 solution for the nonlinear Schrödinger equation, preprint 
  16. [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. MR1097565
  17. [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. 
  18. [16] W.A. Strauss, Existence of solitary waves in hiher dimensions, Comm. Math. Phys., Vol. 55, 1977, p. 149-162. Zbl0356.35028MR454365
  19. [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. Zbl0703.35159
  20. [18] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., Vol. 87, 1983, pp. 567-576. Zbl0527.35023MR691044
  21. [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. Zbl0596.35022MR829596

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.