  In this paper, we consider the Cauchy problem for a hyperbolic equation
$Q(\partial_t,\partial_x)u=0$ of any order $m\geq3$, where $t\geq0$ and
$x\in\mathbb{R}^n$, and $Q=P_m+P_{m-1}+P_{m-2}$ is a sum of homogeneous
hyperbolic polynomials $P_{m-j}$ of order $m-j$. We assume the sufficient and
necessary condition which guarantees the strict stability of the polynomial
$Q(\lambda,i\xi)$, for any $\xi\neq0$. Under this assumption, we derive a
polynomial decay rate for the energy of the problem, in different scenarios of
interlacing of the polynomials $P_{m-j}(\lambda,\xi)$, and we describe the
asymptotic profile of the solution as $t\to\infty$, assuming a moment condition
on the initial data. In order to do this, we study the asymptotic behavior of
the $m$ roots of the full symbol $Q(\lambda,i\xi)$, as $\xi\to0$ and as
$|\xi|\to\infty$. Examples of models to which the results may be applied
include the theory of acoustic waves and the theory of electromagnetic elastic
waves. Also, as an application, we prove the existence of global small data
solutions to the problem with supercritical power nonlinearities of type
$|D^\alpha u|^p$, with $|\alpha|\leq m-2$.

Analysis of PDEs