  We consider a class of time-fractional phase field models including the
Allen-Cahn and Cahn-Hilliard equations. We establish several weighted
positivity results for functionals driven by the Caputo time-fractional
derivative. Several novel criterions are examined for showing the
positive-definiteness of the associated kernel functions. We deduce strict
energy-dissipation for a number of non-local energy functionals, thereby
proving fractional energy dissipation laws.
