Quadratic forms for the 1-D semilinear Schrödinger equation

Kenig, Carlos; Ponce, Gustavo; Vega, Luis

doi:10.1090/S0002-9947-96-01645-5

Quadratic forms for the 1-D semilinear Schrödinger equation
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by Carlos E. Kenig, Gustavo Ponce and Luis Vega PDF

Trans. Amer. Math. Soc. 348 (1996), 3323-3353 Request permission

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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