We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples. Mathematics Subject Classification (2000)

We investigate the long time behaviour of the L 2 -energy of solutions to wave equations with variable speed of propagation. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.

In this paper we derive the following two properties: the first one is a precise representation of WKB solution to the Cauchy problem of a linear wave equation with a variable coefficient with respect to time, and the second one is the global solvability for Kirchhoff equation in some special classes of nonrealanalytic functions, which is proved by grace of the first property.

The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.

We consider the loss of regularity of the solution to the backward Cauchy problem for a second order strictly hyperbolic equation on the time interval [0, T ] with time depending coefficients which have a singularity only at the end point t = 0. The main purpose of this paper is to show that the loss of regularity of the solution on the Gevrey scale can be described by the order of differentiability of the coefficients on (0, T ], the order of singularities of each derivatives as t → 0 and a stabilization condition of the amplitude of oscillations described by an integral on (0, T ). Moreover, we prove the optimality of the conditions for C ∞ coefficients on (0, T ] by constructing a counterexample.