## Differential and Integral Equations

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

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

#### 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