Blowup of solutions for the onedimensional nonlinear Schrödinger equation with critical power nonlinearity
Authors:
Takayoshi Ogawa and Yoshio Tsutsumi
Journal:
Proc. Amer. Math. Soc. 111 (1991), 487496
MSC:
Primary 35B05; Secondary 35Q55
MathSciNet review:
1045145
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We investigate the blowup of solutions in with negative energy for the onedimensional nonlinear Schrödinger equation with critical power nonlinearity: In our result we remove the weight condition , which was always assumed to show the blowup of solutions in the previous papers.
 [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
(89j:35114), http://dx.doi.org/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), 5761.
 [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
(82c:35058), http://dx.doi.org/10.1016/00221236(79)900776
 [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
(57 #842)
 [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
(88f:35133)
 [7]
, Nonlinear Schrödinger equations, preprint.
 [8]
O.
Kavian, A remark on the blowingup of
solutions to the Cauchy problem for nonlinear Schrödinger
equations, Trans. Amer. Math. Soc.
299 (1987), no. 1,
193–203. MR
869407 (88a:35027), http://dx.doi.org/10.1090/S00029947198708694070
 [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
(90i:35249)
 [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
(80k:35056), http://dx.doi.org/10.1016/00221236(78)900733
 [11]
F.
Merle, Limit of the solution of a nonlinear Schrödinger
equation at blowup time, J. Funct. Anal. 84 (1989),
no. 1, 201–214. MR 999497
(90f:35051), http://dx.doi.org/10.1016/00221236(89)901195
 [12]
Frank
Merle, Construction of solutions with exactly 𝑘 blowup
points for the Schrödinger equation with critical nonlinearity,
Comm. Math. Phys. 129 (1990), no. 2, 223–240.
MR
1048692 (91e:35195)
 [13]
Frank
Merle and Yoshio
Tsutsumi, 𝐿² concentration of blowup solutions for
the nonlinear Schrödinger equation with critical power
nonlinearity, J. Differential Equations 84 (1990),
no. 2, 205–214. MR 1047566
(91e:35194), http://dx.doi.org/10.1016/00220396(90)90075Z
 [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
(93h:35193)
 [15]
Hayato
Nawa and Masayoshi
Tsutsumi, On blowup for the pseudoconformally invariant nonlinear
Schrödinger equation, Funkcial. Ekvac. 32
(1989), no. 3, 417–428. MR 1040169
(91e:35193)
 [16]
Takayoshi
Ogawa and Yoshio
Tsutsumi, Blowup of 𝐻¹ solution for the nonlinear
Schrödinger equation, J. Differential Equations
92 (1991), no. 2, 317–330. MR 1120908
(92k:35262), http://dx.doi.org/10.1016/00220396(91)90052B
 [17]
Walter
A. Strauss, Existence of solitary waves in higher dimensions,
Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 0454365
(56 #12616)
 [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, 1989. MR 1032250
(91g:35002)
 [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
(85b:35062), http://dx.doi.org/10.1137/0515028
 [20]
Y. Tsutsumi, Rate of concentration of blowup 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
(84d:35140)
 [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 (87i:35026), http://dx.doi.org/10.1080/03605308608820435
 [23]
, The nonlinear Schrödinger equationSingularity formation, stability and dispersion, The Connection between Infinite and Finite Dimensional Dynamical Systems, Contemp. Math., vol. 99, Amer. Math. Soc., Providence, RI, 1989, pp. 213232.
 [1]
 T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in , Manuscripta Math. 61 (1988), 477498. MR 952091 (89j:35114)
 [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), 5761.
 [4]
 J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations I: The Cauchy problem, II: Scattering theory, general case, J. Funct. Anal. 32 (1979), 171. MR 533219 (82c:35058)
 [5]
 R. T. Glassey, On the blowingup of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18 (1977), 17941797. MR 0460850 (57:842)
 [6]
 T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), 113129. MR 877998 (88f:35133)
 [7]
 , Nonlinear Schrödinger equations, preprint.
 [8]
 O. Kavian, A remark on the blowup of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 299 (1987), 193203. MR 869407 (88a:35027)
 [9]
 B. LeMesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, The focusing singularity of the nonlinear Schrödinger equation, in Direction in partial differential equations (M. G. Crandall, P. H. Rabinowitz, and R. E. Turner, eds.), Academic Press, New York, 1987, pp. 159201. MR 1013838 (90i:35249)
 [10]
 J. E. Lin and W. A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal. 30 (1978), 245263. MR 515228 (80k:35056)
 [11]
 F. Merle, Limit of the solution of the nonlinear Schrödinger equation at the blowup time, J. Funct. Anal. 84 (1989), 201214. MR 999497 (90f:35051)
 [12]
 , Construction of solutions with exactly blowup points for the Schrödinger equation with critical nonlinearity, Comment. Math Phys. 129 (1990), 223240. MR 1048692 (91e:35195)
 [13]
 F. Merle and Y. Tsutsumi, concentration of blowup solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations 84 (1990), 205214. MR 1047566 (91e:35194)
 [14]
 H. Nawa, "Mass" concentration phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity I, II, preprint. MR 1172417 (93h:35193)
 [15]
 H. Nawa and M. Tsutsumi, On blowup for the pseudoconformally invariant nonlinear Schrödinger equation, Funkcialaj Ekvacioj 32 (1989), 417428. MR 1040169 (91e:35193)
 [16]
 T. Ogawa and Y. Tsutsumi, Blowup of solution for the nonlinear Schrödinger equation, J. Differential Equations (to appear). MR 1120908 (92k:35262)
 [17]
 W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149162. MR 0454365 (56:12616)
 [18]
 , Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, no. 73, Amer. Math. Soc., Providence, RI, 1989. MR 1032250 (91g:35002)
 [19]
 M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal. 15 (1984), 357366. MR 731873 (85b:35062)
 [20]
 Y. Tsutsumi, Rate of concentration of blowup solutions for the nonlinear Schrödinger equation with critical power nonlinearity, Nonlinear Anal. (to appear).
 [21]
 M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567576. MR 691044 (84d:35140)
 [22]
 , On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11 (1986), 545565. MR 829596 (87i:35026)
 [23]
 , The nonlinear Schrödinger equationSingularity formation, stability and dispersion, The Connection between Infinite and Finite Dimensional Dynamical Systems, Contemp. Math., vol. 99, Amer. Math. Soc., Providence, RI, 1989, pp. 213232.
Similar Articles
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/S00029939199110451455
PII:
S 00029939(1991)10451455
Article copyright:
© Copyright 1991 American Mathematical Society
