  The numerical integration of phase-field equations is a delicate task which
needs to recover at the discrete level intrinsic properties of the solution
such as energy dissipation and maximum principle. Although the theory of energy
dissipation for classical phase field models is well established, the
corresponding theory for time-fractional phase-field models is still
incomplete. In this article, we study certain nonlocal-in-time energies using
the first-order stabilized semi-implicit L1 scheme. In particular, we will
establish a discrete fractional energy law and a discrete weighted energy law.
The extension for a $(2-{\alpha})$-order L1 scalar auxiliary variable scheme
will be investigated. Moreover, we demonstrate that the energy bound is
preserved for the L1 schemes with nonuniform time steps. Several numerical
experiments are carried to verify our theoretical analysis.
