Remarks on non-homogeneous Cauchy problems with time-dependent generators

In this paper it is studied the well-posedness in several senses for non-homogeneous Cauchy problems where the infinitesimal generator depends on the time parameter. More specifically, we analyze the existence of classical, mild and weak solutions and their relationships. Thanks to uniqueness arguments, mild solutions are proved to satisfy a classical variational formulation. Finally, these results are applied to a thermoelastic plate model where the thermal part is of Cattaneo type and all the physical coefficients depend on time.