On the decay of one-dimensional locally and partially dissipated hyperbolic systems

TL;DR

This paper analyzes the decay rates of linear hyperbolic systems with localized partial dissipation, showing that decay is delayed by the time characteristics spend in undamped regions, under certain stability conditions.

Contribution

It provides a quantitative analysis of decay delays in hyperbolic systems with localized damping, extending classical decay results to systems with space-dependent dissipation.

Findings

Decay rates depend on the time characteristics spend in undamped regions.

The decay delay is proportional to the sum of times characteristics remain undamped.

Classical decay rates are recovered under specific stability conditions.

Abstract

We study the time-asymptotic behavior of linear hyperbolic systems under partial dissipation which is localized in suitable subsets of the domain. More precisely, we recover the classical decay rates of partially dissipative systems satisfying the stability condition (SK) with a time-delay depending only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space-interval and that the system without space-restriction on the dissipation satisfies the stability condition (SK). The former assumption ensures that the time spent by the characteristics of the system in the undamped region is finite and the latter that whenever the damping is active the solutions decay. Our approach consists in reformulating the system into n coupled transport equations and showing that the time-decay estimates are delayed by the sum of the times each characteristics spend in the undamped region.