Quadratic forms for the 1D semilinear Schrödinger equation
Authors:
Carlos E. Kenig, Gustavo Ponce and Luis Vega
Journal:
Trans. Amer. Math. Soc. 348 (1996), 33233353
MSC (1991):
Primary 35K22; Secondary 35P05
MathSciNet review:
1357398
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Abstract: This paper is concerned with 1D quadratic semilinear Schrödinger equations. We study local well posedness in classical Sobolev space of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.
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 T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos 22 Universidade Federal do Rio de Janeiro.
 [CW]
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 [D]
 D. Dix, Nonuniqueness and uniqueness in the initialvalue problem for Burger's equation, SIAM J. Math. Anal. (to appear).
 [F1]
 C. Fefferman, Inequalties for strongly singular convolution operators, Acta Math. 124 (1970), 936. MR 41:2468
 [F2]
 C. Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 4452. MR 47:9160
 [GV1]
 J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979), 171. MR 82c:35057 and MR 82c:35058
 [GV2]
 J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pure Appl. 64 (1985), 363401. MR 87i:35171
 [K1]
 T. Kato, Quasilinear equations of evolutions, with applications to partial differential equations, Lecture Notes in Math, 448, SpringerVerlag, 1975, pp. 2750. MR 53:11252
 [K2]
 T. Kato, On the Cauchy problem for the (generalized) Kortewegde Vries equation, Advances in Math. Supp. Studies, Studies in Applied Math. 8 (1983), 93128. MR 86f:35160
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 [KPV2]
 C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 121. MR 94g:35196
 [KPV3]
 C. E. Kenig, G. Ponce and L. Vega, Bilinear estimates with applications to the KdV equations, J. Amer. Math. Soc. (to appear).
 [KM1]
 S. Klainerman and M. Machedon, Spacetime estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 12211268. MR 94h:35137
 [KM2]
 S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, preprint.
 [L]
 H. Lindblad, A sharp counter example to local existence of low regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993), 503539. MR 94h:35165
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 G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. P.D.E. 18 (1993), 169177. MR 95a:35092
 [S]
 R. Strichartz, Restriction of Fourier transforms to quadratic surface and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705714. MR 58:23577
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 Y. Tsutsumi, solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva. 30 (1987), 115125. MR 89c:35143
 [Z]
 A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189201. MR 52:8788
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Additional Information
Carlos E. Kenig
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email:
cek@math.uchicago.edu
Gustavo Ponce
Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106
Email:
ponce@math.ucsb.edu
Luis Vega
Affiliation:
Departamento de Matematicas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
Email:
MTPVEGOL@lg.ehu.es
DOI:
http://dx.doi.org/10.1090/S0002994796016455
PII:
S 00029947(96)016455
Keywords:
Schrödinger equation,
bilinear estimates,
wellposedness
Received by editor(s):
May 17, 1995
Additional Notes:
C. E. Kenig and G. Ponce were supported by NSF grants. L. Vega was supported by a DGICYT grant.
Article copyright:
© Copyright 1996
American Mathematical Society
