Convergence Rate toward Rarefaction Wave under Periodic Perturbation for Generalized Korteweg-de Vries-Burgers equation

RESUMO

In this paper, a rarefaction wave under space-periodic perturbation for the generalized Korteweg-de Vries-Burgers equation is considered. It is shown that if the initial perturbation around the rarefaction wave is suitably small, then the solution of the system tends to the rarefaction wave. The stability of solution under periodic perturbation is an interesting and important problem since the perturbation keeps oscillating at the far fields. 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 energy method.