Energy-dissipation for time-fractional phase-field equations

<p style='text-indent:20px;'>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 Caputo derivative. Several novel criterions are examined showing positive-definiteness associated kernel functions. deduce strict energy-dissipation number non-local energy functionals, thereby proving fractional dissipation laws.</p>