Hyperbolic-like estimates for higher order equations

The main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time-dependent coefficients. In particular, we estimate the energy in the hyperbolic zone of the extended phase space by means of a function  f ( t ) which depends on the principal part and on the coefficients of the terms of order  m − 1 . Then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space. We call this class of estimates hyperbolic-like since the energy behavior is deeply depending on the hyperbolic structure of the equation. In some cases, these estimates produce a dissipative effect on the energy.