On the asymptotic behavior of the energy for the wave equations with time depending coefficients

Abstract

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)

Energy estimates for wave equations with time dependent propagation speeds in the Gevrey class

“…Clearly, β < 1 holds since α < 1 and m 2; thus β = 1 is not crucial for GEC anymore. Moreover, it is also proved in [3] that GEC is not valid for α > β in general though a ∈ C ∞ ([0, ∞)) satisfies (1.7) for any m ∈ N and (1.8). Consequently, for a ∈ C ∞ ([0, ∞)), we observe that α = β is the critical case for GEC.…”

Long time asymptotics for 2 by 2 hyperbolic systems

“…A stabilization condition (Hypothesis 5) will be introduced. 3. A condition taking account of possible interaction of oscillations of the entries of the matrix A(t) (Hypothesis 3) will be introduced.…”

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

“…which describes an equivalence of the energy with respect to t introduced in [5] as the generalized energy conservation. Here f g for positive functions f, g denotes that there exists a constant C > 1 such that the estimates C −1 g ≤ f ≤ Cg are valid.…”

“…Some non-trivial conditions for (GEC) to the coefficient of (1.1) have been considered in several papers. In particular, we focus to the following conditions from [5], which take account of the regularity of the coefficients:…”

“…A contribution of further smoothness of a(t) than C 2 was studied in [5] with the stabilization condition (1.4). The following theorem is proved:…”