Local well-posedness of the fifth-order KdV-type equations on the half-line

Márcio Cavalcante; Chulkwang Kwak

doi:10.3934/cpaa.2019117

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2019, Volume 18, Issue 5: 2607-2661. Doi: 10.3934/cpaa.2019117

This issue Previous Article On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians Next Article Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials

Local well-posedness of the fifth-order KdV-type equations on the half-line

Márcio Cavalcante^1, and
Chulkwang Kwak^2, ,

1.
Instituto de Matemática, Universidade Federal de Alagoas, Av. Lourival Melo Mota, s/n Tabuleiro do Martins, Maceió, Alagoas, Brazil
2.
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Santiago, Chile and Institute of Pure and Applied Mathematics, Chonbuk National University

^* Corresponding author
^* Corresponding author

Received: August 2018

Revised: January 2019

Published: April 2019

The second author is supported by FONDECYT de Postdoctorado 2017 Proyecto No. 3170067

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Abstract

This paper is a continuation of authors' previous work [6]. We extend the argument [6] to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $ X^{s,b} $ nonlinear estimates for $ b < \frac12 $, which is almost optimal compared to the standard $ X^{s,b} $ nonlinear estimates for $ b > \frac12 $ [8,17]. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.

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Mathematics Subject Classification: 35Q53, 35G31.

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