I have been given the following wave equation:
$$u_{tt}(x,t) = c^2u_{xx}(x,t)-ru_t(x,t)+g(x)\sin(\omega t)$$
where $t >0, x\in (0,l)$ and $g: [0,l] \rightarrow \mathbb{R}$ is in $L^2$. All the parameters are greater zero. Given initial values $u = u_t = 0 \ , \ t=0$ as well as homogeneous boundary conditions: $u(0,t) = u(l,t) = 0$ .I would like to find out the following:
For which values of $\omega$ does the solution grow over time $t$ , when $r = 0$ and for which values $\omega$ when $r > 0$? I am not quite sure how to tackle this problem. In the first case, i.e. $r=0$, I thought I might find an explicit solution but I do not think that this is the best way to solve this, since in the second case I would not be able to find any solution.
I do not necessarily need a complete solution, but any hints or help on how this could be approached would be greatly appreciated!