Let $V,H$ be infinite dimensional separable Hilbert spaces such that $V \hookrightarrow H \hookrightarrow V^*$, where each space is continuously injected in the next one with density. We define:
$$ H(V) := H(a,b ; V, V^*) := \{ u \in L^2(a,b ; V) : u' \in L^2(a,b ; V^*) \} $$
Fix $u_0 \in H$ and $f \in L^2(a,b ; V^*)$. We can set up the problem: find $u \in H(V)$ such that
\begin{aligned} \frac{d}{dt} & \langle u(t), v \rangle_{V^* \times V} + a(u(t),v) = \langle f(t),v \rangle_{V^* \times V} \\ &u(0)=u_0 \end{aligned} for all $v \in V$. Where $a(\cdot , \cdot )$ is a continuous bilinear form (possibly depending on time) on $V \times V$.
I would be grateful if anyone could provide a reference for the study of existance and uniqueness for this type of problems in this abstract framework, maybe just in the easier case $ V \hookrightarrow H$ (with the necessary modifications). Thank you in advance.