  We provide sharp decay estimates in time in the context of Sobolev spaces,
for smooth solutions to the one dimensional Jin-Xin model under the diffusion
scaling, which are uniform with respect to the singular parameter of the
scaling. This provides convergence to the limit nonlinear parabolic equation
both for large time, and for the vanishing singular parameter. The analysis is
performed by means of two main ingredients. First, a crucial change of
variables highlights the dissipative property of the Jin-Xin system, and allows
to observe a faster decay of the dissipative variable with respect to the
conservative one, which is essential in order to close the estimates. Next, the
analysis relies on a deep investigation on the Green function of the linearized
Jin-Xin model, depending on the singular parameter, combined with the Duhamel
formula in order to handle the nonlinear terms.

Analysis of PDEs