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∈V| (Ru)′∈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∋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.