A note on the space-time variational formulation for the wave equation with source term in $L^2(Q)$

We derive a variational formulation for the scalar wave equation in the second-order formulation on bounded Lipschitz domains and homogeneous initial conditions. We investigate a variational framework in a bounded space-time cylinder $Q$ with a new solution space and the test space $L^2(Q)$ for source terms in $L^2(Q)$. Using existence and uniqueness results in $H^1(Q)$, we prove that this variational setting fits the inf-sup theory, including an isomorphism as solution operator. Moreover, we show that the new solution space is not a subspace of $H^2(Q)$. This new uniqueness and solvability result is not only crucial for discretizations using space-time methods, including least-squares approaches, but also important for regularity results and the analysis of related space-time boundary integral equations, which form the basis for space-time boundary element methods.

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