  We investigate the diffusive relaxation limit and the time-asymptotic
stability of the Jin-Xin model toward viscous conservation laws in
$\mathbb{R}^d$ with $d\geq1$. First, we establish uniform regularity estimates
with respect to both the time and the relaxation parameter $\varepsilon>0$, for
initial data in hybrid Besov spaces based on general $L^{p}$-norms. This
uniformity enables us to derive $\mathcal{O}(\varepsilon)$ bounds on the
difference between solutions of the viscous conservation law and its associated
Jin-Xin approximation, thus justifying the strong convergence of the Jin-Xin
hyperbolic relaxation. Furthermore, under an additional condition on the
initial data, for instance, that the low frequencies belong to
$L^{p/2}(\mathbb{R}^{d})$, we show that the $L^{p}(\mathbb{R}^d)$-norm of the
solution to the Jin-Xin model decays at the optimal rate $(1+t)^{-d/{2p}}$
while the $L^{p}(\mathbb{R}^d)$-norm of its difference with the solution of the
associated viscous conservation law decays at the enhanced rate
$\varepsilon(1+t)^{-d/{2p}-1/2}$.
