Degenerate 3-evolution equations in Gevrey classes

We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in \mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t=0$ with finite order. We establish sufficient conditions on the behavior of the lower order coefficients $a_j(t,x)$ $j=1,2$ as $t\to 0^{+}$ and $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ that ensure well-posedness in $L^2(\mathbb{R})$, $H^{\mathrm{\infty }}(\mathbb{R})$ and Gevrey-type spaces.