Decay for the nonlinear KdV equations at critical lengths

[摘要] The nonlinear KdV equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right is considered. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the L2-norm if their initial data belong to a finite dimensional space M. We show that all solutions of the nonlinear system decay to 0 at least with the rate 1/ t1/2 when dimM = 1 or when dimM is even and a specific condition is satisfied, for sufficiently small initial data. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. Consequently, we rediscover all known results by a different approach and obtain new results. We also show that the decay rate is not slower than ln(t + 2)/ t for all critical lengths. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

[发布日期] 2021-09-15

[关键词] KdV equations;Critical lengths;Decay of solutions;Asymptotically stable;Quasi-periodic functions