Let $M$ be a compact Riemannian manifold that is closed. Define $$W(0,T, H^1, L^2) = \{ u \in L^2(0,T;H^1(M)) \mid u_t \in L^2(0,T;L^2(M))\}$$ $$W(0,T, H^1, H^{-1}) = \{ u \in L^2(0,T;H^1(M)) \mid u_t \in L^2(0,T;H^{-1}(M))\}$$ Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?
There must be some result like $C^\infty([0,T]\times M)$ is dense in $W(0,T;H^1, H^{-1})$ but I can't seem to find a reference..