I'm reading P42 on Robert C. Rogers' book "An introduction to partial differential equations" , and I found the way he classifies higher order PDEs a little confusing, especially when it comes to hyperbolic and parabolic ones.
For example, I know $u_{xxxx}-u_{xxyy}+u_{yyyy}=0$ is elliptic since $L^p$ (the principal part) = $\xi^4-\xi^2\eta^2+\eta^4 = 0$ is only possible when $(\xi,\eta)=(0,0)$.
But for this equation $u_{tt}+u_{xxxx}=0$, the principal part $L^p = \xi^4 = 0$ means that $(\xi,\eta)=(0, \eta)$, for all $\eta\in \mathbb{R}$. Does it make this equation parabolic?
Another question: Can we always classify higher order PDEs as parabolic, hyperbolic or elliptic?