  We present a wave packet analysis of a class of possibly degenerate parabolic
equations with variable coefficients. As a consequence, we prove local
wellposedness of the corresponding Cauchy problem in spaces of low regularity,
namely the modulation spaces, assuming a nonlinearity of analytic type. As
another application, we deduce that the corresponding phase space flow
decreases the global wave front set. We also consider the action on spaces of
analytic functions, provided the coefficients are analytic themselves.
