  In this work, we consider a time-fractional Allen-Cahn equation, where the
conventional first order time derivative is replaced by a Caputo fractional
derivative with order $\alpha\in(0,1)$. First, the well-posedness and (limited)
smoothing property are systematically analyzed, by using the maximal $L^p$
regularity of fractional evolution equations and the fractional Gr\"onwall's
inequality. We also show the maximum principle like their conventional
local-in-time counterpart. Precisely, the time-fractional equation preserves
the property that the solution only takes value between the wells of the
double-well potential when the initial data does the same. Second, after
discretizing the fractional derivative by backward Euler convolution
quadrature, we develop several unconditionally solvable and stable time
stepping schemes, i.e., convex splitting scheme, weighted convex splitting
scheme and linear weighted stabilized scheme. Meanwhile, we study the discrete
energy dissipation property (in a weighted average sense), which is important
for gradient flow type models, for the two weighted schemes. Finally, by using
a discrete version of fractional Gr\"onwall's inequality and maximal $\ell^p$
regularity, we prove that the convergence rates of those time-stepping schemes
are $O(\tau^\alpha)$ without any extra regularity assumption on the solution.
We also present extensive numerical results to support our theoretical findings
and to offer new insight on the time-fractional Allen-Cahn dynamics.
