If you want f to take values in $V^*$ rather than $H$, you can do this if you assume more temporal regularity on f. Basically, the idea is to integrate by parts in the term $\int_0^t <u',f>$ in the energy estimate. You will have no trouble finding results of this type in the literature.

If you want f to take values in $V^*$ rather than $H$, you can do this if you assume more temporal regularity on f. Basically, the idea is to integrate by parts in the term $\int_0^t \langle u',f\rangle$ in the energy estimate. You will have no trouble finding results of this type in the literature.