On the Cauchy problem for $D_t^2-D_x(b(t)a(x))D_x$

We consider the Cauchy problem for second order differential operators with two independent variables $P=D_t^2-D_x(b(t)a(x))D_x$. Assume that $b(t)$ is a nonnegative $C^{n,alpha}$ function and $a(x)$ is a nonnegative Gevrey function of order $s>1$ we prove that the Cauchy problem for $P$ is well-posed in the Gevrey class of any order $s<s^{\mathrm{\prime }}<1+(n+alpha)\mathrm{/}2$.