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General boundary value problems of the Korteweg-de Vries equation on a bounded domain

  • * Corresponding author: Bing-Yu Zhang

    * Corresponding author: Bing-Yu Zhang

The paper is dedicated to Jiongmin Yong for his 60th birthday.
The authors thank anonymous referees for many helpful comments, corrections and suggestions

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  • In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval

    subject to the nonhomogeneous boundary conditions,

    $\begin{equation} B_1u = h_1(t), ~~~~~B_2 u = h_2 (t), ~~~~~ B_3 u = h_3 (t) ~~~~~t>0~~~~~(0.2) \end{equation} $

    where

    $ B_i u = \sum\limits_{j = 0}^2 \left(a_{ij} \partial ^j_x u(0, t) + b_{ij}\partial ^j_x u(L, t)\right), ~~~~~i = 1, 2, 3, $

    and $a_{ij}, \ b_{ij}$ ($j = 0, 1, 2$ and $ i = 1, 2, 3$) are real constants. Under some general assumptions imposed on the coefficients $a_{ij}, \ b_{ij}$, the IBVP (0.1)-(0.2) is shown to be locally well-posed in the space $H^s (0, L)$ for any $s \ge 0$ with $φ ∈ H^s (0, L)$ and boundary values $h_j$, $j = 1, 2, 3$, belonging to some appropriate spaces with optimal regularity.

    Mathematics Subject Classification: Primary: 35Q53, 35Q35; Secondary: 53C35.

    Citation:

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