  We are concerned with quasilinear symmetrizable partially dissipative
hyperbolic systems in the whole space $\mathbb{R}^d$ with $d\geq2$. Following
our recent work [10] dedicated to the one-dimensional case, we establish the
existence of global strong solutions and decay estimates in the critical
regularity setting whenever the system under consideration satisfies the
so-called (SK) (for Shizuta-Kawashima) condition. Our results in particular
apply to the compressible Euler system with damping in the velocity equation.
Compared to the papers by Kawashima and Xu [27, 28] devoted to similar issues,
our use of hybrid Besov norms with different regularity exponents in low and
high frequency enable us to pinpoint optimal smallness conditions for global
well-posedness and to get more accurate information on the qualitative
properties of the constructed solutions. A great part of our analysis relies on
the study of a Lyapunov functional in the spirit of that of Beauchard and
Zuazua in [2]. Exhibiting a damped mode with faster time decay than the whole
solution also plays a key role.
