Here is a wave equation $u_{tt} - \Delta u = \delta(x)e^{-i\omega t} $ and I need to solve it using Duhamel's rule.
The auxiliary function I get is $U_{tt} - \Delta U=0,~U(x,s,s)=0,~U_{t}(x,s,s)=\delta(x)e^{-i\omega s}$ where $U(x,t,s)$ is the solution for $t\geq s$.
By treating it with spherical mean method, I have $U=\frac{\partial}{\partial t}(tF(x,t))+tG(x,t)$ where $F(x,t), G(x,t)$ are the spherical mean for $U(x,s,s),~U_{t}(x,s,s)$ respectively.
But here I get stuck since $F(x,t), G(x,t)$ are both zero. The former is because $U(x,s,s)=0$ and the latter is because integrating for $U_{t}(x,s,s)=\delta(x)e^{-i\omega s}$ on the spherical surface but not the entire sphere also get zero. As far as I know, integral of $\delta(x)$ is $1$ for region including $0$, and is $0$ otherwise.
Any idea how to solve this?