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In this paper, a rarefaction wave under space-periodic perturbation for the 3times 3 rate-type viscoelastic system is considered. It is shown that if theinitial perturbation around the rarefaction wave is suitably small, then thesolution of the rate-type viscoelastic system tends to the rarefaction wave.The stability of solutions under periodic perturbations is an interesting andimportant problem since the perturbation keeps oscillating at the far fields.That is, the perturbation is not integral in space. The key of proof is toconstruct a suitable ansatz carrying the same oscillation as the solution. Thenwe can find cancellations between solutions and ansatz such that theperturbation belongs to some Sobolev space. The nonlinear stability can beobtained by the weighted energy method.

Stability of Rarefaction Waves Under Periodic Perturbation for A Rate-Type Viscoelastic System