2020/12/16 by Nguyen, Hoai-Minh · 2 citations
#35B40 #35C20 #35Q53 #93B05 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2012.08792
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 L2-norm if their initial data belong to a finite dimensional subspace \M. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate 1/ t1/2 when dim \M = 1 or when dim \M 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 dim \M = 1 or for the smallest critical length L with dim \M = 2 by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than ln (t + 2) / t for all critical lengths.