The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand :
Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For example, $\Phi = S(\mathbb R^n)$ is the space of Schwartz functions and $\Phi' = S'(\mathbb R^n)$ is the space of tempered distributions.
Let $A_t$, $t \in [0,T]$, be a family of continuous linear operators from $\Phi$ to $\Phi$ such that for all test functions $\phi \in \Phi$ : $$t \mapsto A_t\phi \in C^0([0,T], \Phi)$$
Let $A_t^*$ be the adjoint of $A_t$, i.e. $A_t^* : \Phi' \rightarrow \Phi'$ is the continuous linear operator defined as $$\langle A_t^* f, \phi \rangle := \langle f, A_t \phi \rangle, \ f \in \Phi', \ \phi \in \Phi$$
Assume that the Cauchy problem
\begin{equation} \begin{cases} \partial_t u(t) &= A_t u(t), \ t \in [0,T] \\ u(t_0) &= u_0 \end{cases}\tag{1} \end{equation}
admits a unique solution $u(t) \in C^1([0,T], \Phi)$ (differentiability in a Fréchet space) for all $0 \leq t_0 \leq T$, $u_0 \in \Phi$. Furthermore, assume that for all fixed $t \in [0,T]$, $u(t)$ depends continuously on $u_0$.
Then the dual problem \begin{equation} \begin{cases} \partial_t u(t) &= -A_t^* u(t), \ t \in [0,T] \\ u(0) &= u_0 \end{cases}\tag{2} \end{equation}
admits a unique solution $u \in C^1([0,T], \Phi')$ (meaning that the derivative $u'(t)$ converges in the weak topology and defines a continuous function of $t$) for all $u_0 \in \Phi'$. Moreover, for all fixed $t \in [0,T]$, the solution $u(t)$ depends continuously on $u_0$.
Precisely, if $Q_{t_0}^t : \Phi \rightarrow \Phi$ is the map that sends $u_0 \in \Phi$ to the unique solution $u(t)$ of $(1)$, then $(Q_{t}^0)^* : \Phi' \rightarrow \Phi'$ is the map that sends $u_0 \in \Phi'$ to the unique solution $u(t)$ of $(2)$.
My question is the following : Does this result generalize to evolution equation of the form
$$P(\partial_t) u(t) = A_t u(t), \ t \in [0,T]$$
with similar assumptions and adequate boundary conditions ? Here, $P \in \mathbb C[x]$.
What if we consider the simpler case where $P(\partial_t) = \partial_t^k$ ?
A reference is welcomed as well.
EDIT :
For the simpler case, the Cauchy problem $$\partial_t^k u(t) = A_t u(t), \ t \in [0,T]$$ in the space $\Phi$ is equivalent to a problem of the form $(1)$ in the space $\Phi^k$ by considering $$v(t) = (u(t), \partial_t u(t), ..., \partial_t^{k-1} u(t))$$
