  We present an abstract framework for treating the theory of well-posedness of
solutions to abstract parabolic partial differential equations on evolving
Hilbert spaces. This theory is applicable to variational formulations of PDEs
on evolving spatial domains including moving hypersurfaces. We formulate an
appropriate time derivative on evolving spaces called the material derivative
and define a weak material derivative in analogy with the usual time derivative
in fixed domain problems; our setting is abstract and not restricted to
evolving domains or surfaces. Then we show well-posedness to a certain class of
parabolic PDEs under some assumptions on the parabolic operator and the data.
