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We consider the Cauchy problem for second order differential operators withtwo independent variables $P=D_t^2-D_x(b(t)a(x))D_x$. Assume that $b(t)$ is anonnegative $C^{n,alpha}$ function and $a(x)$ is a nonnegative Gevrey functionof order $s>1$ we prove that the Cauchy problem for $P$ is well-posed in theGevrey class of any order $s<s'<1+(n+alpha)/2$.

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