Let $V\subset H\subset V^*$ a Hilbert triple and consider a 2nd order evolution equation of the form $$u''(t)+Au(t) = f(t)\quad \text{ in }\ L^2(0,T;V^*),$$ where $f\in\ L^2(0,T;H)$.

**Can we let $f\in L^2(0,T;V^*)$?**

This question is a special case ($A(t)=A$) of Regularity of solution to a hyperbolic pde. There, the answer says

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.

I think by integration by parts it is meant $$\int_0^t \big\langle f(s),v'(s)\big\rangle_{V^*,V}\,\mathrm{d}s=\big(f(t),v(t)\big)_H-\big(f(0),v(0)\big)_H-\int_0^t \big\langle v(s),f'(s)\big\rangle_{V^*,V} \,\mathrm{d}s,$$ but the right hand side does not make sense unless $f'(s)\in V$, and I am confused.

**Where can I find this type of result?**