The remarkable effectiveness of time-dependent damping terms for second order evolution equations

TL;DR

This paper investigates how time-dependent damping coefficients can be optimized to maximize decay rates in second order evolution equations, revealing pulsating coefficients as most effective.

Contribution

It introduces a novel approach to designing time-dependent damping terms that enhance decay rates, especially using pulsating coefficients, across various types of second order equations.

Findings

Pulsating damping coefficients outperform constant and uniformly large coefficients.

Optimal damping involves alternating large and small values over time.

The approach applies to ODEs, systems, and hyperbolic PDEs.

Abstract

We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover that constant coefficients do not achieve the goal, as well as time-dependent coefficients that are too big. On the contrary, pulsating coefficients which alternate big and small values in a suitable way prove to be more effective. Our theory applies to ordinary differential equations, systems of ordinary differential equations, and partial differential equations of hyperbolic type.