Let $\Omega$ be a bounded domain and define $$W(0,T) = \{u \in L^2(0,T;H^1_0(\Omega)) : u_t \in L^2(0,T;H^{-1}(\Omega))\}$$ where $u_t$ means the weak temporal derivative.
I am looking for a result like
A space like $C_c^\infty((0,T)\times\Omega)$ or $C^\infty_c([0,T]\times \Omega)$ is dense in $W(0,T)$.
For example $C^\infty([0,T]; H^1_0(\Omega))$ is known to be dense. But I need smoothness on the product domain. Does anyone have a reference?
The set $C_c^\infty( (0,T)\times \Omega)$ cannot be dense: The evaluation $u \mapsto u(0,\cdot)$ is continuous from $W(0,T)$ to $L^2(\Omega)$. Hence any element $u$ in the closure of $C_c^\infty( (0,T)\times \Omega)$ in $W(0,T)$ satisfies $u(0)=u(T)=0$.
The set $C_c^\infty([0,T] \times\Omega)$ is dense: Take $u\in C^\infty([0,T];H^1_0(\Omega))$. The space $H^1_0$ is separable, so let $(e_n)$ be a complete orthonormal system.
Then $$ u(t) = \sum_n \langle u(t), e_n \rangle e_n. $$ Now approximate each $e_n$ by $\phi_n \in C_c^\infty(\Omega)$ such that $\| e_n - \phi_n \|_{H^1_0(\Omega)} < \epsilon 2^{-n}$. Define $$ u_\epsilon(t) = \sum_n \langle u(t), e_n \rangle \phi_n. $$ Then $$ \|u(t) - u_\epsilon(t)\|_{H^1} \le \epsilon \sum_n \langle u(t), e_n \rangle 2^{-n} \le \epsilon \|u(t)\|_{H^1}. $$ Now we replace $u_\epsilon$ by $u_N:=\sum_{n=1}^N \langle u(t), e_n \rangle \phi_n$, where $u_n \to u_\epsilon$ in $L^\infty(0,T;H^1)$. Choose $N$ so large such that $\|u_\epsilon - u_N\|_{L^\infty(0,T;H^1)}\le \|u_\epsilon - u\|_{L^\infty(0,T;H^1)}$. And $u_N$ is a good approximation of $u$. In addition, as $u_N$ is a finite sum, we have that $u_N \in C^\infty([0,T] \times \Omega)$. Moreover the support of $u_N$ with respect to $x$ is a compact set of $\Omega$, i.e., the closure of $\{x: \ u(x,t) \ne 0 \text{ for some }t\}$ is compact in $\Omega$. So that $u_N \in C_c^\infty([0,T] \times \Omega)$.
Similar estimates can be made for the time derivative, so that $u_N$ is close to $u$ with respect to the $W(0,T)$ norm.