Nonlinear hyperbolic Cauchy problems in Gevrey classes.

Well-posedness of the Cauchy problem in the Gevrey classes for quasi-linear equations of constant multiplicity is stated. More precisely, let $P$ be a hyperbolic equation with $C^\beta$ Hölder continuous coefficients with respect to time, and let $r$ be the largest multiplicity of the characteristics of $P$. Then the Cauchy problem is well-posed in the Gevrey classes of index smaller than $\min(\frac {r}{r-\beta},1+\beta)$. For $\beta=1$ and for linear $P$, the result goes back to the classical theory of perturbation of hyperbolic equations. For non-linear $P$, this is an improvement of related results by K. Kajitani