I am interested in regularity results for solutions to 2nd order evolution equations in the shape of $$ u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\ u(0) = u_0 \text{ in } H, u'(0) = u_1\text{ in } V^*, $$ where $S=(0,T)$ with $0<T<\infty$ and $V\hookrightarrow H \hookrightarrow V^*$ denotes a Gelfand Triple. Standard literature, like Dautray Lions, Wloka, Evans, and others, shows that a unique solution $u\in L^2(S;V)\cap H^1(S;H)\cap H^2(S;V^*)$ exists for $$ A\in W^{1,\infty}(S;\mathcal{L}(V;V^*)), \quad u_0\in V, u_1\in H, \quad f\in L^2(S;H)\text{ or }f\in W^{1,2}(S;V^*). $$ Wloka then also shows an abstract regularity result. Assuming $A':V\to H$, a very restrictive condition, as well as suitable regularity of given data, he proves existence of a unique solution $u\in H^1(S;V)\cap H^2(S;H)\cap H^3(S;V^*)$. But when dealing with a PDE where $A$ represents $-\text{div}(a\nabla u)$ with time-dependent $a$, this restriction on $A'$ is not matched since this operator maps to $V^*$ instead of $H$.
Does anyone know of a reference to a result where $A':V\to H$ is not assumed?
Thank you very much in advance, sgr.
