Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal symbol $a(x,\xi)$. This problem is discussed in a number of sources (Hormander v.iii, Taylor's $\Psi$DO, etc. ).

Let $S(t,s)$ be the propagator from $s$ to time $t$, then when $f=0$ one has $$ \operatorname{WF}(S(t,0)u_0) = \chi_t \operatorname{WF}(u_0) $$ where $\chi_t$ is the flow generated by the Hamiltonian $H_a$.

Now take $f \in C^0([0,T], H^s(\mathbb{R}^n))$, $u_0 = 0$ (for simplicity), then the equation is well-posed and the solution is given by: $$ u(t,x) = \int_0^t S(t,s) f(s) d s$$

**My questions:**

What is the wavefront set of $u$ as a distribution in $(0,T)\times\mathbb{R}^n$?

I'm missing references on the question.

My feeling is that maybe seeing integration as the push-forward of the projection $\pi(t,s,x) = (t,x)$ might work. But then I guess I should see $S(t,s)$ as an FIO in $(t,s,x)$ which sounds strange and probably there is a more elementary derivation