Distribution of energy of solutions of the wave equation on singular spaces of constant curvature and on a homogeneous tree

• Published in: Russian journal of mathematical physics Vol. 23; no. 4; pp. 536 - 550
• Main Author: Tsvetkova, A. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.10.2016; Springer Nature B.V
• Subjects: Cauchy problems; Curvature; Energy distribution; Euclidean geometry; Euclidean space; Infinity; Mathematical analysis; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; Wave equations; Wave power
• ISSN: 1061-9208; 1555-6638
• DOI: 10.1134/S1061920816040099

In the paper, the Cauchy problem for the wave equation on singular spaces of constant curvature and on an infinite homogeneous tree is studied. Two singular spaces are considered: the first one consists of a three-dimensional Euclidean space to which a ray is glued, and the other is formed by two three-dimensional Euclidean spaces joined by a segment. The solution of the Cauchy problem for the wave equation on these objects is described and the behavior of the energy of a wave as time tends to infinity is studied. The Cauchy problem for the wave equation on an infinite homogeneous tree is also considered, where the matching conditions for the Laplace operator at the vertices are chosen in the form generalizing the Kirchhoff conditions. The spectrum of such an operator is found, and the solution of the Cauchy problem for the wave equation is described. The behavior of wave energy as time tends to infinity is also studied.