On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

Abstract

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as C k -type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.

Time-dependent propagation speed vs strong damping for degenerate linear hyperbolic equations

“…We refer to [7] for the history of the problem and a short survey of some related literature (see also [1,4,5,9] and the references quoted therein for analogous models with competition between damping and time-dependent propagation speed). Here we just recall the main results that are more relevant to our presentation.…”