## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 50, Number 2 (2010), 365-401.

### A sufficient condition for well-posedness for systems with time-dependent coefficients

#### Abstract

We consider linear, smooth, hyperbolic systems with time-dependent coefficients and size $N$. We give a condition sufficient for the well-posedness of the Cauchy Problem in some Gevrey classes. We present some Levi conditions to improve the Gevrey index of well-posedness for the scalar equation of order $N$, using the transformation in [DAS] and following the technique introduced in [CT]. By using this result and adding some assumptions on the form of the first-order term, we can improve the well-posedness for systems. A similar condition has been studied in [DAT] for systems with size $3$.

#### Article information

**Source**

Kyoto J. Math., Volume 50, Number 2 (2010), 365-401.

**Dates**

First available in Project Euclid: 7 May 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1273236820

**Digital Object Identifier**

doi:10.1215/0023608X-2009-017

**Mathematical Reviews number (MathSciNet)**

MR2666662

**Zentralblatt MATH identifier**

1203.35163

**Subjects**

Primary: 35L45: Initial value problems for first-order hyperbolic systems

#### Citation

D’Abbicco, Marcello. A sufficient condition for well-posedness for systems with time-dependent coefficients. Kyoto J. Math. 50 (2010), no. 2, 365--401. doi:10.1215/0023608X-2009-017. https://projecteuclid.org/euclid.kjm/1273236820