Is L^2(0,T;H^1(\Omega)) compactly embedded in L^2(0,T;L^2(\Omega))? I know by Rellich-Kondrachov theorem, $H^1(\Omega)$ is compactly embedded $L^2(\Omega)$. Also, $L^2(0,T;H^1(\Omega))$ is continuously embedded in $L^2(0,T;L^2(\Omega))$. However, is this embedding compact? Is there a reference related to it?

By considering the set of constant functions $\Omega \ni x \mapsto c$ (which belong to $H^1$ and $L^2$) show that both of your spaces "contain a copy of $L^2(0,T)$". Since the embedding $L^2(0,T) \hookrightarrow L^2(0,T)$ is not compact, your embedding cannot be compact either.
