An evolution inclusion of subdifferential type with time dependent subdifferentials and a multivalued perturbation is considered in a separable Hilbert space. The perturbation has closed unbounded values. We also consider the inclusion with the values of the perturbation being convexified (convexified inclusion). The notion of a regular solution for the unbounded convexified inclusion is introduced. Any solution of the convexified inclusion with a bounded perturbation is such a solution. The theorems on existence and relaxation for regular solutions of the unbounded convexified inclusion are proved. In contrast to the known results of this kind, we do not suppose that the convex function whose subdifferential appears in the inclusion has the compactness property. Additionally, instead of making the traditional assumption for such problems of Lipschitz continuity of the perturbation in the phase variable in the Hausdorff metric, we use a more natural notion of (rho - H) Lipschitzness for mappings with unbounded values. An example of unbounded convexified inclusion, every solution of which is regular, is given.