  We study, in the periodic setting, the well-posedness of the Cauchy problem
associated to the operator $P(t, D_{x}, D_{t}) = D_{t} - a_{2}(t) \Delta_{x} +
\sum_{j = 1}^{N} a_{1, j}(t) D_{x_{j}} + a_{0}(t)$, with $T> 0$, $t \in [0, T]$
and $a_{2}, a_{1,1}, \ldots, a_{1, N}, a_{0} \in C \left([0, T]; \mathbb{C}
\right)$. Using Fourier analysis techniques, we obtain a complete
characterization for the well-posedness of a class of degenerate initial-value
problems in the Sobolev, Smooth, Gevrey and Real-analytic frameworks.
