Let $\Omega$ be open and bounded.
Is there anything nice I can say about the space $C^1([0,T]\times \Omega)$ and its inclusion in some Bochner like spaces?
If $f \in C^1([0,T]\times \Omega)$ then it is not true that $f \in C^0([0,T]; C^1(\Omega))$. And I don't think it's even true that $f \in C^1([0,T]; C^0(\Omega)).$ How about if we say $f$ (and some of its derivatives) are bounded for all $x,t$?
My reason for asking this is I want to know: If $f \in C^1([0,T]\times \Omega)$ and $g \in C^1_c((0,T); W^{s,p}(\Omega))$ then does the product rule apply: $(fg)' = f'g + fg'$ (the $'$ means the time derivative)