  There exists a well defined energy for classical phase-field equations under
which the dissipation law is satisfied, i.e., the energy is non-increasing with
respect to time. However, it is not clear how to extend the energy definition
to time-fractional phase-field equations so that the corresponding dissipation
law is still satisfied. In this work, we will try to settle this problem for
phase-field equations with Caputo time-fractional derivative, by defining a
nonlocal energy as an averaging of the classical energy with a time-dependent
weight function. As the governing equation exhibits both nonlocal and nonlinear
behavior, the dissipation analysis is challenging. To deal with this, we
propose a new theorem on judging the positive definiteness of a symmetric
function, that is derived from a special Cholesky decomposition. Then, the
nonlocal energy is proved to be dissipative under a simple restriction of the
weight function. Within the same framework, the time fractional derivative of
classical energy for time-fractional phase-field models can be proved to be
always nonpositive.
