  The initial inverse problem of finding solutions and their initial values ($t
= 0$) appearing in a general class of fractional reaction-diffusion equations
from the knowledge of solutions at the final time ($t = T$). Our work focuses
on the existence and regularity of mild solutions in two cases: \begin{itemize}
\item[--] The first case: The nonlinearity is globally Lipschitz and uniformly
bounded which plays important roles in PDE theories, and especially in
numerical analysis. \item[--] The second case: The nonlinearity is locally
critical which widely arises from the Navier-Stokes, Schr\"odinger, Burgers,
Allen-Cahn, Ginzburg-Landau equations, etc. \end{itemize} Our solutions are
local-in-time and are derived via fixed point arguments in suitable functional
spaces. The key idea is to combine the theories of Mittag-Leffler functions and
fractional Sobolev embeddings. To firm the effectiveness of our methods, we
finally apply our main results to time fractional Navier-Stokes and Allen-Cahn
equations.
