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 https://mathoverflow.net/questions/191800/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 $\int_0^t <u',f>$ 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 
$$\int_0^t <f(s),v'(s)>_{V^*,V}ds=(f(t),v(t))_H-(f(0),v(0))_H-\int_0^t <v(s),f'(s)>_{V^*,V} ds,$$
but the right hand side does not make sense unless $f'(s)\in V$, and I am confused. I am actually having trouble finding results of this type in the literature.  

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