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We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the -norm if their initial data belong to a finite dimensional subspace . In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate when or when is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for or for the smallest critical length with by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than for all critical lengths.
Annalisa Buffa, Pablo Antolin Sanchez, Giuliano Guarino