  We develop a variational approach in order to study the qualitative
properties of non-autonomous parabolic equations. Based on the method of
product integrals, we discuss long-time behavior, invariance properties, and
ultracontractivity of evolution families in Hilbert space. Our main results
give sufficient conditions for the heat kernel of the evolution family to
satisfy Gaussian-type bounds. Along the way, we study examples of
non-autonomous equations on graphs, metric graphs, and domains.
