Necessary and sufficient conditions for the existence of a classical solution of the mixed problem for the wave equation on a graph

Abstract

We study a mixed problem for the wave equation with integrable potential on the simplest geometric graph consisting of two ring edges that touch at a point. We use a new approach in the Fourier method to obtain necessary and sufficient conditions for the existence of a classical solution. We do not use refined asymptotic formulas for the eigenvalues and any information on the eigenfunctions. The solution is represented by a rapidly convergent series.