Hyperbolic partial differential equation

Definition

Let $U \subseteq \R^n$ be open, let $T > 0$, and write the variables in $U \times (0, T)$ as $(x, t)$. Consider a linear PDE in $U \times (0, T)$ of the following form. $$\partial_t^2 u - \sum_{i,j=1}^{n} a_{ij}(x, t) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x, t) \, \partial_i u + b_0(x, t) \, \partial_t u + c(x, t) \, u = f(x, t).$$ Here all coefficients are real and $a_{ij} = a_{ji}$. The equation is a second-order linear hyperbolic equation if its spatial part is uniformly elliptic. That is, there is $\theta > 0$ with $\sum_{i,j} a_{ij}(x, t) \xi_i \xi_j \ge \theta \abs{\xi}^2$ for all $(x, t)$ and all $\xi \in \R^n$.

More generally, let $P$ be a linear operator of order $m$ in the variables $(x, t)$, with principal symbol $\sigma_m$. It is hyperbolic with respect to the time direction at a point $y$ if both conditions hold:

1. $\sigma_m(y, (0, \dots, 0, 1)) \ne 0$;
2. for every $\xi \in \R^n$ the polynomial $\tau \mapsto \sigma_m(y, (\xi, \tau))$ has only real roots.

It is strictly hyperbolic if, in addition, these $m$ roots are distinct for every $\xi \ne 0$.