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](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 $\int\_0^t \langle u',f \rangle$ 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?**



Solution of hyperbolic equations with $V^*$ data
Chapter 9 in Volume 1 of Lions/Magenes [1] treats this case, even for nonautonomous operators. One essentially gets (somewhat as expected?) a regularity shift just in the spatial components, so the solution $u$ will satisfy $(u,u') \in C([0,T];H \times V^\*)$. (This is provided the initial values for $(u,u')$ are in $H \times V^\*$, of course.) For nonautonomous operators $A$ you will however have to assume additional time regularity in order to recover uniqueness of solutions. This circumvents the lack of an energy equality which one would normally use, I guess.


[1] *Lions, J. L.; Magenes, E.*, Non-homogeneous boundary value problems and applications. Vol. I. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften. Band 181. Berlin-Heidelberg-New York: Springer-Verlag. XVI,357 p. (1972). [ZBL0223.35039](https://zbmath.org/?q=an:0223.35039).


