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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?

Lousouf
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    See here – Artem Sep 07 '21 at 12:44
  • @Artem thank you! this clarifies a lot! But I'm still confused about the second equation in my post. If I follow what is said in that article, then for no $\tau$ will the polynomial in s, $(\xi + s\tau)^4=0$ have 4 distinct real roots. So it's neither elliptic nor strictly hyperbolic, right? – Lousouf Sep 07 '21 at 15:14
  • How did you obtain the principle part for the first equation? I was able to do it for part a in the same book. I guess where I'm stuck is obtaining the matrix form – Jaider Sep 17 '22 at 21:26
  • In the text, they explain how to obtain the principal part for second order PDE. I'm so confused on how to do it for higher order PDE, the formulation I can't find on google – Jaider Sep 18 '22 at 00:40

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