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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 , \e …

A pretty extensive reference on Huygens' principle is [G] Günther, Paul MR 946226 Huygens’ principle and hyperbolic equations, Perspectives in Mathematics ISBN: 0-12-307330-8. First, some definition …

I believe that a zero eigenvalue of $A(u)$ implies that the vector field $\partial_t$ is a characteristic direction. For instance, the equation $u_s + u_y = 0$ takes the form $u_t=0$ after switching t …

To my knowledge, the principal symbol of a non-linear differential operator is not discussed very often. When I have seen it discussed, the definition basically coincided with your approach 1. For exa …

I will add a pessimistic answer. You are right that Choque-Bruhat's (and any related local-in-time) existence result only guarantees that the solution metric exists and is sufficiently regular (includ …

For normally hyperbolic operators (those whose principal symbol is the same as for the wave operator, but possibly acting on a vector bundle, the theory of fundamental solutions/Green functions (as di …

Let me start by writing $1/U$ instead of $U$. The case when $U$ or $1/U$ vanishes (or $V$ vanishes) would require special handling of course. Then your constraint is $$ \frac{\omega_u}{\omega_v} = \fr …

I think you are asking about the possibility of satisfying the desired boundary conditions by each solution of the Helmholtz equation in the product form $X(x) Y(y)$, where $X(x)$ an $Y(y)$ were obtai …

Section 4 of the following paper considers in some detail the 2D wave equation ($\partial_x\partial_y f = 0$ in your coordinates; not exactly the same but closely related) on compact domains with smoo …

It's difficult for me to check the details at the moment, but the book Non-Homogeneous Boundary Value Problems and Applications by Lions and Magenes might also be helpful, perhaps more so for elliptic …

Consider a determined linear system of differential order $k$ of the form $\sigma^{i_1\cdots i_k}_{ab}(x) \partial_{i_1} \cdots \partial_{i_k} u^b(x) + l.o.t = 0$. The coefficients $\sigma^{i_1\cdots …

Have a look at Ringström's The Cauchy Problem in General Relativity (EMS 2009). He spends several chapters building up the analytical material of the kind that you are asking about.