We study the propagation of high frequency oscillations for one dimensional semi-linear hyperbolic systems with small parabolic perturbations. We obtain a new degenerate parabolic system for the profile, and valid an asymptotic development in the spirit of Joly, Métivier and Rauch.
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Stéphane Junca, Geometric optics with critical vanishing viscosity for one-dimensional semilinear initial value problems. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 549–566DOI 10.4171/RMI/546