Energy Analysis of Some Singular Initial Value Problems

This paper deals with a problem of conservation of energy in a singular initial value problem. These kind of problems appear in the study of radially symmetric elliptic and hyperbolic boundary value problems. The interesting question that arises is how to analyze the oscillation of the solutions, count the number of nodes and estimate their location. In order to conduct the analysis we need to have a very good understanding of the energy of the corresponding solutions and prove results that show how the system contains enough energy to provide oscillation of the solution. We also measure the rate of energy decay in every oscillation that occurs. These results are crucial in solving a number of problems appearing in geometry, engineering, physics, etc.