this is my first post in this forum, so please do not hesitate to comment on any possible improvements for any future post regarding form, e.g. Thanks in advance. 

I have a question concerning 2nd order evolution equation of the form $u''(t)+A(t)u(t) = f(t)$ in $L^(0,T;V^*)$, where $f\in\ L^2(0,T;H)$ holds. Under what assumptions is it possible, to guarantee a unique solution for a more general $f\in L^2(0,T;V^*)$. I wrote down an exact setting following Wloka in his book *partial differential equations:*

Be $t\in [0,T]$ with $0<T<\infty$ and let the bilinear form $a(t;u,v)$ be symmetric and continuous for all $t$, i.e.,

$|a(t;u,v)|\le const.\ \|u\|_V\|\ v\|_V,\quad \forall\ u,v\in V$.

We assume for all $u,v\in V$ the mapping $t\mapsto a(t;u,v)$ to be continuously differentiable and V-coercive, i.e., 

$\exists\ k_1,k_2>0:\ \forall\ t\in [0,T],\ v\in V:\ a(t;v,v) + k_1|v|_H^2 \ge k_2\|v\|_V^2$

Suppose 

$V:=W^{1,2}(\Omega),\ H:=L^{2}(\Omega),\ f\in L^2(0,T;H)$

and

$u_0\in V,\ u_1\in H$

then the second order evolution equation

$\langle u''(t),v\rangle +a(t;u(t),v)=\langle f(t),v\rangle\quad in\ V^*\quad f.a.a.\ t\in (0,T)$  

with initial values 

$u(0)=u_0\in V,\ u'(0)=u_1\in H$

possesses a unique solution 

$u\in L^2(0,T;V),\ u'\in L^2(0,T;H),\ u''\in L^2(0,T;V^*)$. (cf. Zeidler nonlinear functional analysis and its applications IIa, Wloka (as above)). 

I found a possibility in *Dautray and Lions: Mathematical analysis and numerical methods for science and technology volume 5*, but there an additional term on the left-hand side is necessary, with a coercive Operator acting on $u'$. These and comparable results give me the impression, that my question has to be answered with *no*. But since I am fairly new to this topic, I am thankfull for every suggestion for regularity theory on hyperbolic equations and general ideas to my problem. 

Best regards,
Simon.