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

We study the time-asymptotic behavior of linear hyperbolic systems subject to partial dissipation that is localized in suitable subsets of the domain. Specifically, we recover the classical decay rates of partially dissipative systems that satisfy the stability condition (SK), with a time-delay that depends 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 ensures that the solutions decay whenever the damping is active. Our approach consists of reformulating the system into $n$ coupled transport equations and showing that the time-decay estimates are delayed by the sum of the times that each characteristic spends in the undamped region.