Journal of Mathematics of Kyoto University

The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation

Hiroyuki Chihara

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Abstract

We present the local and the global existence theorems for the elliptic-hyperbolic Davey-Stewartson equation which does not allows the classical energy estimates. To overcome this difficulty, we make use of the smoothing property of linear Schrödinger type equations which was obtained by S. Doi. Then under the smallness condition to $L^{2}$-norm of the initial data, we get the local solution. Moreover we show the global existence of small amplitude solutions by the a priori estimates for which the null gauge condition of Y. Tsutsumi plays an important role.

Article information

Source
J. Math. Kyoto Univ., Volume 39, Number 1 (1999), 41-66.

Dates
First available in Project Euclid: 17 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1250517953

Digital Object Identifier
doi:10.1215/kjm/1250517953

Mathematical Reviews number (MathSciNet)
MR1684172

Zentralblatt MATH identifier
0947.35036

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35M10: Equations of mixed type

Citation

Chihara, Hiroyuki. The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation. J. Math. Kyoto Univ. 39 (1999), no. 1, 41--66. doi:10.1215/kjm/1250517953. https://projecteuclid.org/euclid.kjm/1250517953


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