  Many physical phenomena may be modelled by first order hyperbolic equations
with degenerate dissipative or diffusive terms. This is the case for example in
gas dynamics, where the mass is conserved during the evolution, but the
momentum balance includes a diffusion (viscosity) or damping (relaxation) term,
or, in numerical simulations, of conservation laws by relaxation schemes. Such
so-called partially dissipative systems have been first pointed out by S.K.
Godunov in a short note in Russian in 1961. Much later, in 1984, S. Kawashima
highlighted in his PhD thesis a simple criterion ensuring the existence of
global strong solutions in the vicinity of a linearly stable constant state.
This criterion has been revisited in a number of research works. In particular,
K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing
a Lyapunov functional allowing to refine Kawashima's results and to establish
global existence results in some situations that were not covered before. These
notes originate essentially from the PhD thesis of T. Crin-Barat that was
initially motivated by an earlier observation of the author in a Chapter of the
handbook coedited by Y. Giga and A. Novotn{\'y}. Our main aim is to adapt the
method of Beauchard and Zuazua to a class of symmetrizable quasilinear
hyperbolic systems (containing the compressible Euler equations), in a critical
regularity setting that allows to keep track of the dependence with respect to
e.g. the relaxation parameter. Compared to Beauchard and Zuazua's work, we
exhibit a 'damped mode' that will have a key role in the construction of global
solutions with critical regularity, in the proof of optimal time-decay
estimates and, last but not least, in the study of the strong relaxation limit.
For simplicity, we here focus on a simple class of partially dissipative
systems, but the overall strategy is rather flexible, and adaptable to much
more involved situations.
