  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.
