  We aim to classify the long-time behavior of the solution to a free boundary
problem with monostable reaction term in space-time periodic media. Such a
model may be used to describe the spreading of a new or invasive species, with
the free boundary representing the expanding front. In time-periodic and space
homogeneous environment, as well as in space-periodic and time autonomous
environment, such a problem has been studied recently in \cite{dgp, dl}. In
both cases, a spreading-vanishing dichotomy has been established, and when
spreading happens, the asymptotic spreading speed is proved to exist by making
use of the corresponding semi-wave solutions. The approaches in \cite{dgp, dl}
seem difficult to apply to the current situation where the environment is
periodic in both space and time. Here we take a different approach, based on
the methods developed by Weinberger \cite{w1, w2} and others
\cite{fyz,lyz,lz1,lz2,lui}, which yield the existence of the spreading speed
without using traveling wave solutions.
  In Part 1 of this work, we establish the existence and uniqueness of
classical solutions for the free boundary problem with continuous initial data,
extending the existing theory which was established only for $C^2$ initial
data. This will enable us to develop Weinberger's method in Part 2 to determine
the spreading speed without knowing a priori the existence of the corresponding
semi-wave solutions. In Part 1 here, we also establish a spreading-vanishing
dichotomy.
