Let $u \in H^1((0,T)\times S)$ be the unique solution of $$u_{tt} + \Delta u =0$$ $$u|_{t=0}= u_0$$ $$u|_{t=T}=0$$ where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without boundary (eg. unit sphere). The weak formulation satisfied is $$\int_0^T\int_S u_tv_t + \nabla u \nabla v = 0$$ for each $v \in H^1((0,T)\times S)$ such that $v|_{t=0}=v|_{t=T}=0$.
Is it possible to obtain the following bound: $$\lVert u \rVert_{L^2((0,T)\times S)} \leq C\lVert{u_0}\rVert_{L^2(S)}?$$ So I want a bound on the $L^2$ norm involving only the $L^2$ of the initial data, not the full $H^{\frac 12}$ norm.
