Stability of rarefaction waves under periodic perturbation for a rate-type viscoelastic system. (English) Zbl 07915182
Summary: In this paper, a rarefaction wave under space-periodic perturbation for the \(3 \times 3\) rate-type viscoelastic system is considered. It is shown that if the initial perturbation around the rarefaction wave is suitably small, then the solution of the rate-type viscoelastic system tends to the rarefaction wave. The stability of solutions under periodic perturbation is an interesting and important problem since the perturbation keeps oscillating at the far fields. That is, the perturbation is not integrable in space. The key of proof is to construct a suitable ansatz carrying the same oscillation as the solution. Then we can find cancellations between solutions and ansatz such that the perturbation belongs to some Sobolev space. The nonlinear stability can be obtained by the weighted energy method.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35L60 | First-order nonlinear hyperbolic equations |
Keywords:
Cauchy problem; rate-type viscoelastic system; periodic perturbations; asymptotic behavior; time decay rate; rarefaction waveReferences:
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