To get some feeling for the problem, we make the following simplifying assumptions:
$B$ is independent of $x$, say $B \in S(\overline{\mathbb R}_+)$.
$A$ has its spectrum contained in $\overline{\mathbb H}_-$ and admits a (holomorphic) functional calculus.
There is a (holomorphic) function $H\colon \sigma(A)\to{\mathbb C}$ such that $\bigl(i\tau - \lambda\,\hat{B}(\tau)\bigr)\bigr|_{\,\tau=H(\lambda)} =0$ for $\lambda\in\sigma(A)$, where $\hat{B}(\tau)=\int_0^\infty e^{-it\tau}B(t)\,dt$ is the Fourier transform of $B$.
Then it is readily seen that $$ u(t) = e^{itH(A)} u_0 $$ is a solution of the original problem.
There are several difficult points with this approach, the two most severe ones being:
The range of $i\tau\,/\,\hat{B}(\tau)$ might miss parts of the spectrum of $A$. This puts restrictions on the initial value $u_0$.
There might be several choices for the function $H$. This leads to a non-uniqueness of the solution $u$.
Still, if the assumptions are made in a way that this approach works, then the operator $e^{itH(A)}$ allows one to study the propagation of singularities and, moreover, there is a parametrix construction when $B$ also depends on $x$.