  Here we develop a method for investigating global strong solutions of
partially dissipative hyperbolic systems in the critical regularity setting.
Compared to the recent works by Kawashima and Xu, we use hybrid Besov spaces
with different regularity exponent in low and high frequency. This allows to
consider more general data and to track the exact dependency on the dissipation
parameter for the solution. Our approach enables us to go beyond the L^2
framework in the treatment of the low frequencies of the solution, which is
totally new, to the best of our knowledge. Focus is on the one-dimensional
setting (the multi-dimensional case will be considered in a forthcoming paper)
and, for expository purpose, the first part of the paper is devoted to a toy
model that may be seen as a simplification of the compressible Euler system
with damping. More elaborated systems (including the compressible Euler system
with general increasing pressure law) are considered at the end of the paper.
