Blow-up of solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity

Authors:
Takayoshi Ogawa and Yoshio Tsutsumi

Journal:
Proc. Amer. Math. Soc. **111** (1991), 487-496

MSC:
Primary 35B05; Secondary 35Q55

MathSciNet review:
1045145

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the blow-up of solutions in with negative energy for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity:

**[1]**Thierry Cazenave and Fred B. Weissler,*The Cauchy problem for the nonlinear Schrödinger equation in 𝐻¹*, Manuscripta Math.**61**(1988), no. 4, 477–494. MR**952091**, 10.1007/BF01258601**[2]**-,*The structure of solutions to the pseudoconformally invariant nonlinear Schrödinger equation*, Proc. Roy. Sci. Edinburgh (to appear).**[3]**L. M. Degtyarev, V. E. Zakharov, and L. I. Rudakov,*Two examples of Langmuir wave collapse*, Soviet Phys. JETP**41**(1975), 57-61.**[4]**J. Ginibre and G. Velo,*On a class of nonlinear Schrödinger equations. II. Scattering theory, general case*, J. Funct. Anal.**32**(1979), no. 1, 33–71. MR**533219**, 10.1016/0022-1236(79)90077-6**[5]**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**0460850****[6]**Tosio Kato,*On nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré Phys. Théor.**46**(1987), no. 1, 113–129 (English, with French summary). MR**877998****[7]**-,*Nonlinear Schrödinger equations*, preprint.**[8]**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**869407**, 10.1090/S0002-9947-1987-0869407-0**[9]**B. LeMesurier, G. Papanicolaou, C. Sulem, and P.-L. Sulem,*The focusing singularity of the nonlinear Schrödinger equation*, Directions in partial differential equations (Madison, WI, 1985) Publ. Math. Res. Center Univ. Wisconsin, vol. 54, Academic Press, Boston, MA, 1987, pp. 159–201. MR**1013838****[10]**Jeng Eng Lin and Walter A. Strauss,*Decay and scattering of solutions of a nonlinear Schrödinger equation*, J. Funct. Anal.**30**(1978), no. 2, 245–263. MR**515228**, 10.1016/0022-1236(78)90073-3**[11]**F. Merle,*Limit of the solution of a nonlinear Schrödinger equation at blow-up time*, J. Funct. Anal.**84**(1989), no. 1, 201–214. MR**999497**, 10.1016/0022-1236(89)90119-5**[12]**Frank Merle,*Construction of solutions with exactly 𝑘 blow-up points for the Schrödinger equation with critical nonlinearity*, Comm. Math. Phys.**129**(1990), no. 2, 223–240. MR**1048692****[13]**Frank Merle and Yoshio Tsutsumi,*𝐿² concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity*, J. Differential Equations**84**(1990), no. 2, 205–214. MR**1047566**, 10.1016/0022-0396(90)90075-Z**[14]**Hayato Nawa,*“Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity*, Funkcial. Ekvac.**35**(1992), no. 1, 1–18. MR**1172417****[15]**Hayato Nawa and Masayoshi Tsutsumi,*On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation*, Funkcial. Ekvac.**32**(1989), no. 3, 417–428. MR**1040169****[16]**Takayoshi Ogawa and Yoshio Tsutsumi,*Blow-up of 𝐻¹ solution for the nonlinear Schrödinger equation*, J. Differential Equations**92**(1991), no. 2, 317–330. MR**1120908**, 10.1016/0022-0396(91)90052-B**[17]**Walter A. Strauss,*Existence of solitary waves in higher dimensions*, Comm. Math. Phys.**55**(1977), no. 2, 149–162. MR**0454365****[18]**Walter A. Strauss,*Nonlinear wave equations*, CBMS Regional Conference Series in Mathematics, vol. 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR**1032250****[19]**Masayoshi Tsutsumi,*Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations*, SIAM J. Math. Anal.**15**(1984), no. 2, 357–366. MR**731873**, 10.1137/0515028**[20]**Y. Tsutsumi,*Rate of**concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity*, Nonlinear Anal. (to appear).**[21]**Michael I. Weinstein,*Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys.**87**(1982/83), no. 4, 567–576. MR**691044****[22]**Michael I. Weinstein,*On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations*, Comm. Partial Differential Equations**11**(1986), no. 5, 545–565. MR**829596**, 10.1080/03605308608820435**[23]**-,*The nonlinear Schrödinger equation--Singularity formation, stability and dispersion*, The Connection between Infinite and Finite Dimensional Dynamical Systems, Contemp. Math., vol. 99, Amer. Math. Soc., Providence, RI, 1989, pp. 213-232.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
35B05,
35Q55

Retrieve articles in all journals with MSC: 35B05, 35Q55

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1045145-5

Article copyright:
© Copyright 1991
American Mathematical Society