# Generalized wave equation

I am interested in understanding as much as I can about the following partial differential equation, which is a generalization of the 1D wave equation:

$$\frac{\partial^2 u(x,t)}{\partial t^2} + \zeta(t)\frac{\partial u(x,t)}{\partial t}= \alpha(t)\frac{\partial^2 u(x,t)}{\partial x^2} + \beta(t)\frac{\partial u(x,t)}{\partial x} + \gamma(t)u(x,t) + \alpha(t)f(x),\quad x\in(-\infty,\infty),\quad t\geq 0$$

where $\zeta(t)\geq 0$, $\beta(t)\geq 0$, and $\alpha(t)\geq 0$. This equation has initial data $u(x,0)$ and velocity $\partial_t u(x,0)$. Here are a few examples:

1. When $\zeta(t) = \beta(t) = \gamma(t) = f(t) = 0$, it is the standard 1D wave equation. Thus, the influence of the initial data travels at a speed $\alpha(t)$.

2. When $\zeta(t) = \gamma(t) = f(t) = 0$, it is the 1D Klein-Gordon equation. Thus, the influence of the initial data travels at a speed $\leq\alpha(t)$. Should I be thinking about dispersion relations in this case?

3. When $\beta(t) = \gamma(t) = f(t) = 0$, it is the 1D wave equation with dampening. Should I be thinking about lack of energy conservation in this case?

4. When $\zeta(t) = \beta(t) = \gamma(t) = 0$ and $f(x) = \delta_0(x)$ ($\delta_0 =$ delta function at origin), it is the 1D wave equation with plucking at $x = 0$?

I would like to understand properties of the solution for any choice of $\alpha, \beta,\gamma,\zeta,f$. What tools are there to help me? When is there a conservation of energy statement? Thank you very much in advance.

When $\beta(t)=k\alpha(t)$ , where $k$ is a constant, the PDE is separable:
Let $u(x,t)=X(x)T(t)$ ,
Then $X(x)T''(t)+\zeta(t)X(x)T'(t)=\alpha(t)X''(x)T(t)+k\alpha(t)X'(x)T(t)+\gamma(t)X(x)T(t)$
$X(x)T''(t)+\zeta(t)X(x)T'(t)-\gamma(t)X(x)T(t)=\alpha(t)X''(x)T(t)+k\alpha(t)X'(x)T(t)$
$X(x)(T''(t)+\zeta(t)T'(t)-\gamma(t)T(t))=\alpha(t)T(t)(X''(x)+kX'(x))$
$\dfrac{T''(t)+\zeta(t)T'(t)-\gamma(t)T(t)}{\alpha(t)T(t)}=\dfrac{X''(x)+kX'(x)}{X(x)}$