Non-homogeneous wave equation on a cone

Abstract

The wave equation $\left({\partial }_{tt}-c^2{\Delta }_x\right)u(x,t)=e^{-t}f(x,t)$ is shown to have a unique solution if $u$ and its partial derivatives in $x$ are in $L^2(e^{-t})$ on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.