The existence of the solution of the wave equation on graphs
Yong Lin, Yuanyuan Xie · arXiv (Cornell University) · 2019
Let $G=(V, E)$ be a finite weighted graph, and $Ω\subseteq V$ be a domain such that $Ω^\circ eq\emptyset$. 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)-Δ_Ωu(t,x)=f(t,x),\qquad&&(t,x)\in[0,\infty)\times Ω^\circ,\\ &u(0,x)=g(x),\qquad&& x\inΩ^\circ,\\ &\partial_tu(0,x)=h(x),\qquad&& x\inΩ^\circ,\\ &u(t,x)=0,\qquad&&(t,x)\in[0,\infty)\times\partial Ω, \end{aligned} \right. \end{equation*} where $Δ_Ω$ denotes the Dirichlet Laplacian on $Ω^\circ$. Using Rothe's method, we prove that the above wave equation has a unique solution.