Identification problems for singular integro-differential equations of parabolic type I

We recover unknown kernels, depending on time only, in linear singular first-order integro-differential Cauchy problems in Banach spaces. Singular means here that the integro-differential equation is not in normal form neither can it be reduced to such a form. For this class of problems we prove local and global in time existence and uniqueness theorems strictly related to the regularity results proved in [4] for the direct problem. Moreover, we give several applications to explicit singular partial integro-differential equations of parabolic type.