See the book of A. Bensoussane et al. for different notions of solutions in section: 3 Nonhomogeneous linear evolution equations, especially Proposition 3.1 and 3.2 where it is proved that some of these notions coincide.

>A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2007.