On the energy decay estimates for the weak dissipative wave equations with oscillating coefficient
Fumihiko Hirosawa, Daichi Nakajima
TL;DR
The paper analyzes energy decay for the dissipative wave equation with weak, oscillatory damping $b(t)=\frac{m}{1+t}+\delta(t)$, using a time-frequency framework inspired by Ghisi-Gobbino. It derives sharp decay bounds for the energy density by partitioning the $(t,\xi)$-space into hyperbolic and dissipative zones and employing a polar-coordinate representation of the solution; oscillations are encoded through a periodically varying $\sigma_0(\eta)$ and a phase $\eta(t)$, with the decay rate governed by the parameter $m$ and the stabilization/regularity of the perturbations. A secondary, Gevrey-regularity result shows that initial data in a Gevrey class can yield decay without loss even when oscillations would otherwise degrade the rate. The work clarifies the interplay between oscillatory weak dissipation, zone-specific energy estimates, and initial-data smoothness, connecting to broader themes in time-dependent propagation and energy conservation.
Abstract
It is known that the asymptotic behavior of time-dependent dissipative coefficient in the Cauchy problem of dissipative wave equation dominates the energy decay estimate. In particular, it is important to study the case where the dissipative coefficient behave like $1/t$ as $t$ goes to infinity, which is called weak dissipation, because its order is close to the critical case of decay and non-decay. In this case, an oscillating perturbation of weak dissipation can give a crucial effect on the energy decay estimate, but the analysis is very difficult compared to the case without oscillations. In this paper, we develop a method recently introduce by Ghisi-Gobbino that has contributed to a precise analysis for dissipative wave equations with oscillating weak dissipation, and consider the effect of the oscillations, which is more general and close to the critical case. Furthermore, we study the effect of the smoothness of the initial data on the energy decay estimates.