Differential and Integral Equations

Some new well-posedness results for the Klein-Gordon-Schrödinger system

Hartmut Pecher

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Abstract

We consider the Cauchy problem for the 2D and 3D Klein-Gordon Schrödinger system. In 2D we show local well posedness for Schrödinger data in $H^s$ and wave data in $H^{\sigma} \times H^{\sigma -1}$ for $s=-1/4 \, +$ and $\sigma = -1/2$, whereas ill posedness holds for $s < - 1/4$ or $\sigma < -1/2$, and global well-posedness for $s\ge 0$ and $s-\frac{1}{2} \le \sigma < s+ \frac{3}{2}$. In 3D we show global well posedness for $s \ge 0$, $ s - \frac{1}{2} < \sigma \le s+1$. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru [2], and Bejenaru and Herr [3] for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schrödinger system by Colliander, Holmer and Tzirakis [5].

Article information

Source
Differential Integral Equations, Volume 25, Number 1/2 (2012), 117-142.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012829

Mathematical Reviews number (MathSciNet)
MR2906550

Zentralblatt MATH identifier
1249.35309

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35L70: Nonlinear second-order hyperbolic equations

Citation

Pecher, Hartmut. Some new well-posedness results for the Klein-Gordon-Schrödinger system. Differential Integral Equations 25 (2012), no. 1/2, 117--142. https://projecteuclid.org/euclid.die/1356012829


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