The Cauchy problem for a class of linear degenerate evolution equation on the torus

We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator $P(t,D_x,D_t)=D_t-a_2(t){\mathrm{\Delta }}_x+{\sum }_{j=1}^N{a}_{1,j}(t)D_{x_j}+a_0(t)$, with $T>0$, $t\in [0,T]$ and $a_2,a_{1,1},\dots ,a_{1,N},a_0\in C([0,T];\mathbb{C})$. 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.