Asymptotics of the Solution of a Wave Equation with Radially Symmetric Velocity on the Simplest Decorated Graph with Arbitrary Boundary Conditions at the Gluing Poi

摘要

We consider the Cauchy problem for a wave equation with variable velocity on the simplest decorated graph obtained by gluing a ray to R~3, with initial conditions localized on the ray. For the wave operator to be self-adjoint, we impose certain boundary conditions at the gluing point. This paper describes the asymptotic expansion of the solution of the problem under consideration for arbitrary boundary conditions at the gluing point under the assumption that the velocity on R3 is radially symmetric. Also we study the distribution of the energy of the wave as the small parameter tends to zero, which depends on the boundary conditions.