= Solution
A <solution operator for a nonautonomous evolution equation> is an <evolution family> $U(t,s)$ satisfying
$$
U(s,s)=I,
\qquad
U(t,r)U(r,s)=U(t,s),
$$
and, on a suitable common domain $\mathcal D$,
$$
\partial_tU(t,s)u=A(t)U(t,s)u,
\qquad
\partial_sU(t,s)u=-U(t,s)A(s)u.
$$
One applicable nonautonomous generation theorem is the following. Suppose $\mathcal D$ is a dense linear subspace of $H$, each $A(t)$ has domain $\mathcal D$, the family is a <stable family of semigroup generators> with constants $M,0$, and $t\mapsto A(t)$ is continuously differentiable as a map from $[0,T]$ to $\mathcal B(\mathcal D,H)$, where $\mathcal D$ carries one of the uniformly equivalent <graph norms>. Then there is a unique <evolution family> such that:
* $(t,s)\mapsto U(t,s)u$ is continuous for every $u\in H$ and $\|U(t,s)\|\leq M$;
* $U(t,s)\mathcal D\subseteq\mathcal D$, with a uniform bound on $U(t,s)$ as an operator on $\mathcal D$;
* for $u\in\mathcal D$, both displayed differential equations hold in $H$.
For the uniform partition $t_j=s+j(t-s)/N$, the <frozen-generator product approximation> is
$$
U_N(t,s)
=e^{(t_N-t_{N-1})A(t_{N-1})}
\cdots e^{(t_1-t_0)A(t_0)}.
$$
As $N\to\infty$, $U_N(t,s)u\to U(t,s)u$ in the norm of $H$ for every $u\in H$, uniformly for $(t,s)$ in the compact time triangle $0\leq s\leq t\leq T$. This is convergence in the <strong operator topology>, rather than convergence in the <operator norm>.
It remains to verify the second differential equation. The <evolution family> law gives, for $h>0$,
$$
U(t,s+h)u-U(t,s)u
=U(t,s+h)\bigl[u-U(s+h,s)u\bigr].
$$
Divide by $h$. Since $u\in\mathcal D$,
$$
\frac{U(s+h,s)u-u}{h}\longrightarrow A(s)u,
$$
while <strong continuity> gives $U(t,s+h)A(s)u\to U(t,s)A(s)u$. Therefore
$$
\boxed{\partial_sU(t,s)u=-U(t,s)A(s)u}.
$$
The left derivative follows in the same way, so $s\mapsto U(t,s)u$ is differentiable.
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