We investigate the well-posedness of the generalized wave equation posed on the space-time cylinder Q over a bounded Lipschitz domain. By employing the Hodge-Laplace operator over k-forms, we utilize a unifying framework that elegantly recovers both the standard scalar wave equation (for 0-forms) and Maxwell's equations (for 1-forms under suitable gauges). Our variational formulation relies on specialized space-time tensor spaces, constructed via the Hilbert tensor products of temporal Sobolev spaces and spatial spaces of k-forms.
As our primary result, we establish the existence and uniqueness of solutions for square-integrable source terms j in L^2 in both space and time. The proof employs a Fourier series ansatz, combining spatial eigenforms and temporal basis functions to derive a rigorous stability estimate for the solution. Furthermore, we investigate the theoretical boundaries of this functional setting. We demonstrate that the space of admissible source terms cannot be extended to the dual space of the test functions, as the requisite inf-sup condition from the Banach-Nečas-Babuška theorem definitively fails in this broader context.