In this talk, we are interested in understanding the smoothing and the localizing properties of two classes of evolution equations. The first class is associated with the accretive quadratic operators (which are non-selfadjoint differential operators), for which partial hypoellipticity phenomena occur, which will be explained. The second class is associated with the fractional anisotropic Shubin operators (defined as the fractional power of differential operators), whose smoothing and localizing properties are subject to an uncertainty principle, as we will see. This a work partly joint with J. Bernier (LMJL).