The existence of the solution of the wave equation on graphs

Let $G=(V, E)$ be a finite weighted graph, and $\Omega\subseteq V$ be a domain such that $\Omega^\circ\neq\emptyset$. In this paper, we study the following initial boundary problem for the non-homogenous wave equation \begin{equation*} \left\{ \begin{aligned} &\partial_t^2 u(t,x)-\Delta_\Omega u(t,x)=f(t,x),\qquad&&(t,x)\in[0,\infty)\times \Omega^\circ,\\ &u(0,x)=g(x),\qquad&& x\in\Omega^\circ,\\ &\partial_tu(0,x)=h(x),\qquad&& x\in\Omega^\circ,\\ &u(t,x)=0,\qquad&&(t,x)\in[0,\infty)\times\partial \Omega, \end{aligned} \right. \end{equation*} where $\Delta_\Omega$ denotes the Dirichlet Laplacian on $\Omega^\circ$. Using Rothe's method, we prove that the above wave equation has a unique solution.