The model example of wave equation with oscillating scale-invariant damping
Marina Ghisi, Massimo Gobbino
TL;DR
This work analyzes a damped wave equation with scale-invariant damping and an oscillatory component that is integrable but not absolutely integrable. Using Fourier analysis and a polar-coordinate formulation, the authors reduce the problem to a family of ODEs and establish a resonance mechanism between the damping oscillation and elastic frequencies, leading to slower energy decay: $\mathcal{D}(t) \ge \frac{C_1(t_0,m,r)}{t^{m-r/2}}$. The method hinges on a detailed study of the phase $\theta(t)$, showing $\theta(t)=t+\varphi(t)$ with $\varphi(t)\to 0$, and a careful analysis of oscillatory integrals that govern the energy evolution. The results extend to abstract wave equations with $b(t)=\frac{\beta(t)}{t}$ when $\beta$ is periodic and resonates with spectral frequencies, and the paper outlines several generalizations, including other frequencies, phases, and non-resonant perturbations, pointing to broad implications for how oscillatory damping shapes long-time dynamics in dispersive systems.
Abstract
We analyze a simple example of wave equation with a time-dependent damping term, whose coefficient decays at infinity at the scale-invariant rate and includes an oscillatory component that is integrable but not absolutely integrable. We show that the oscillations in the damping coefficient induce a resonance effect with a fundamental solution of the elastic term, altering the energy decay rate of solutions. In particular, some solutions exhibit slower decay compared to the case without the oscillatory component. Our proof relies on Fourier analysis and a representation of solutions in polar coordinates, reducing the problem to a detailed study of the asymptotic behavior of solutions to a family of ordinary differential equations and suitable oscillatory integrals.