# Generalized wave equation

I asked this question here: https://math.stackexchange.com/questions/1160134/generalized-wave-equation but did not get any response. I hope it is more suitable on mathoverflow.

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.

Using your coordinates, we can define a Lorentzian metric $g = - dt^2 + \frac{1}{\alpha(t)} dx^2$. Then your equation takes the form $$\square u + v^i \partial_i u + \gamma u = h ,$$ where $\square = \frac{1}{\sqrt{-\det g}} \partial_i \sqrt{-\det g} g^{ij} \partial_j$ is the (d'Alambert) wave operator for the metric $g$, $v^i \partial_i = -(\zeta(t) - \alpha'(t)/2\alpha(t))\partial_t + \beta(t)\partial_x$ is a vector field, $\gamma = \gamma(t)$ and $h = \alpha(t) f(t)$. If the Lorentzian metric $g$ and the coefficients $v$, $\gamma$ and $h$ are arbitrary functions, you have the most general form of a linear inhomogeneous second order normally hyperbolic equation. You have simply chosen very special kinds of coefficients that depend only on time.