Abstract: By using the method of energy analysis, firstly, we considered the wave equation defined on a three-dimensional semi-infinite cylinder. When the spatial variable approached infinity, we proved that the solution of the equation either grew exponentially or decayed exponentially. Secondly, we considered the wave equation defined on the exterior region of the sphere, and proved the alternative result of its solution with radius. Finally, we proved that the alternative theorem was still valid for nonlinear elastic equations.

Key words: wave equation, energy analysis, exterior region, elastic equation