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

Abstract

MSC:The total energy of the wave equation is conserved with respect to time if the propagation speed is a constant, but this is not true in general for time dependent propagation speeds. Indeed, it is considered in Hirosawa (2007) [3] that the following properties of the propagation speed are crucial for the estimates of the total energy: oscillating speed, difference from the mean, and the smoothness in C m category. The main purpose of this paper is to derive a benefit of a further smoothness of the propagation speed in the Gevrey class for the energy estimates.

On second order weakly hyperbolic equations and the ultradifferentiable classes

“…Then our main theorem is given as follows: (1.9) Remark 2.2. The energy estimate of solution to (1.1) for a 0 > 0 and T = ∞ with the coefficient in the Gevrey classes is studied in [14]. This result is corresponding to the case that λ(t) = 1, …”

On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions

“…We also see that lim m→∞ β m = α, hence (1.5) provides almost optimal condition as m → ∞. In [9], a limit case m → ∞ is studied to consider a(t) in the Gevrey class, which is a subclass of C ∞ . Table 1: Taking into account Theorem 1.1 and the remarks above, the main purposes of this article are extending Theorem 1.1 from the following points of view:…”

Response Solutions for Wave Equations with Variable Wave Speed and Periodic Forcing

“…More precisely, they described the Gevrey and C ∞ well-posedness of (1.2) with respect to Hölder and Log-Lipschitz continuity of a(•), respectively. In [20,21], Hirosawa established the energy estimate for the d-dimensional wave equation u tt − a(t) u = 0.…”