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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quadratic forms for the 1-D semilinear Schrödinger equation


Authors: Carlos E. Kenig, Gustavo Ponce and Luis Vega
Journal: Trans. Amer. Math. Soc. 348 (1996), 3323-3353
MSC (1991): Primary 35K22; Secondary 35P05
MathSciNet review: 1357398
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Abstract: This paper is concerned with 1-D quadratic semilinear
Schrödinger equations. We study local well posedness in classical Sobolev space $H^s$ of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of $s$ 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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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/S0002-9947-96-01645-5
PII: S 0002-9947(96)01645-5
Keywords: Schrödinger equation, bilinear estimates, well-posedness
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