Generalized wave equation

Question

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.

Answer 1

Using your coordinates, we can define a Lorentzian metric $g = - dt^2 + \frac{1}{\alpha(t)} dx^2$. Then your equation takes the form \begin{equation} \square u + v^i \partial_i u + \gamma u = h , \end{equation} 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.

These kinds of equations have been extensively studied and you can find lots of information about them, for instance, in the thorough monographs

• Bär, Ginoux & Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization (EMS, 2007).
• Waldmann, Geometric Wave Equations (2012).