I am dealing with the problem

\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\ \partial_{\nu} u &= 0 &\text{ on } \partial \Omega \times (0,T)\\ u(x,0) &= u_0(x) &\text{ in } \Omega\end{align}

on some bounded domain $\Omega$ with $a_1 \leq a(x,t) \leq a_2$ for constants $a_1,a_2 >0$.

We set $A(t)u = \nabla \cdot (a(\cdot,t) \nabla u)$ on some $D(A(t))$ that is independent from $t$ and includes the condition $\partial_{\nu} u = 0$ on $\partial \Omega$. For example $D(A(t)) = \{\varphi \in W^{2,q} (\Omega): \,\partial_{\nu} \varphi = 0 \}$?

If $a$ would not depend on $t$ the solution would (for appropriate $a$ and $f$) be given by \begin{align} u(t) = e^{tA}u_0 + \int_0^te^{(t-s)A} f(s) \,ds. \end{align}

Is there some similar result for this particular $A$ depending on $t$?