Asymptotics of higher order hyperbolic equations with one or two dissipative lower order terms

Abstract

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 $Q(λ,iξ)$, for any $ξ\neq0$. Under this assumption, we derive a polynomial decay rate for the energy of the problem, in different scenarios of interlacing of the polynomials $P_{m-j}(λ,ξ)$, 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 $Q(λ,iξ)$, as $ξ\to0$ and as $|ξ|\to\infty$. 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 $|D^αu|^p$, with $|α|\leq m-2$.