  A new framework to obtain time-decay estimates for partially dissipative
hyperbolic systems set on the real line is developed. Under the classical
Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank
condition in control theory, the solutions of these systems decay exponentially
in time for high frequencies and polynomially for low ones. This allows to
derive a sharp description of the space-time decay of solutions for large time.
However, such analysis relies heavily on the use of the Fourier transform that
we avoid here, developing the "physical space version" of the hyperbolic
hypocoercivity approach introduced by Beauchard and Zuazua, to prove new
asymptotic results in the linear and nonlinear settings. The new physical space
version of the hyperbolic hypocoercivity approach allows to recover the natural
heat-like time-decay of solutions under sharp rank conditions, without
employing Fourier analysis or $L^1$ assumptions on the initial data. Taking
advantage of this Fourier-free framework, we establish new enhanced time-decay
estimates for initial data belonging to weighted Sobolev spaces. These results
are then applied to the nonlinear compressible Euler equations with linear
damping. We also prove the logarithmic stability of the nonlinearly damped
$p$-system.
