Zero Relaxation Limit to Centered Rarefaction Waves for a Rate-Type Viscoelastic System

We study a rate-type viscoelastic system proposed by I. Suliciu (1990, Internat. J. Engrg. Sci.28, 827–841), which is a 3×3 hyperbolic system with relaxation. As the relaxation time tends to zero, this system convergences to the well-known p-system formally. In the case where the initial data are the Riemann data such that the corresponding solutions of the p-system are centered rarefaction waves, we show that if the wave strength is suitably small, then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero, except for an initial layer. The jump discontinuities in the solutions are decaying exponentially fast as time tends to infinity.