Existence results for a class of evolution equations of mixed type

We give an existence result for the evolution equation (Ru)' + Au = f in the space W = {u is an element of V\ (Ru)' is an element of V'}where V is a Banach space and R is a non-invertible operator (the equation may be partially elliptic and partially parabolic, both forward and backward) and we study the "Cauchy-Dirichlet" problem associated to this equation (indeed also for the inclusion (Ru)' + Au There Exists f). We also investigate continuous and compact embeddings of W and regularity in time of the solution. At the end we give some examples of different R.