Localized Asymptotic Solution of a Variable-Velocity Wave Equation on the Simplest Decorated Graph with Initial Conditions on a Surface

A variable-velocity wave equation is studied on the simplest decorated graph, i.e., the topological space obtained by attaching a ray to $\mathbb R^3$. The Cauchy problem with initial conditions localized on Euclidean space is considered. The leading term of an asymptotic solution of the problem under consideration as the parameter characterizing the size of the source tends to zero is described by using the construction of the Maslov canonical operator. It is assumed that the point on $\mathbb R^3$ at which the ray is attached is not a singular point of the wavefront.