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Boundary-value problem for the Korteweg-de Vries-Burgers type equation

  • Nakao Hayashi
  • Elena I. Kaikina
  • H. Francisco Ruiz Paredes

Abstract.

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation on half-line¶¶\((x,t)\in {{\bf R}^{+}}\times {{\bf R} ^{+}}$,\cr u(x,0)=u_{0}(x),$ x\in {{\bf R}^{+}}$,\hspace*{8pc}(1)\cr u(0,t)=0,$ t\in {{\bf R}^{+}}\)¶¶We prove that if the initial data \( u_{0}\in {\bf X} \), and the norm \( \Vert u_{0}\Vert _{{\bf X}} \) is sufficiently small, where \( {\bf X}=\{\varphi \in {\bf L}^{1}\cap {\bf H}^{1};\Vert \varphi \Vert _{{\bf X}}=\Vert \varphi \Vert _{{\bf L}^{1}}+\Vert \varphi \Vert _{{\bf H}^{1}}\)<\( \infty \} \), then there exists a unique solution \( u\in {\bf C}([0,\infty );{\bf H}^{1}) \) of the initial-boundary value problem (1), where H k is the Sobolev space with norm \( \Vert \phi \Vert _{{\bf H}^{k}}=\Vert (1-\partial _{x}^{2})^{\frac{k}{2}}\phi \Vert _{{\bf L}^{2}}. \) We also find the large time asymptotics of the solutions under the condition \( x^{1+\delta }u\in {\bf L}^{1}\cap {\bf L}^{2} \) with \( \delta \in (0,1). \) More pricesely, we prove¶¶\( u(x,t)=\frac{A}{t}e^{-\frac{x^{2}}{4t}}\frac{x}{2\sqrt{t}}+O\Bigg(\min \left( 1,\frac{x}{2\sqrt{t}}\right) t^{-1-\frac{\delta}{2}}\Bigg), \)¶¶where A will be defined below in Theorem 2.

Key words: Boundary value problems, Korteweg-de Vries-Burgers equation. 

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Copyright information

© Birkhäuser Verlag, Basel, 2001

Authors and Affiliations

  • Nakao Hayashi
    • 1
  • Elena I. Kaikina
    • 2
  • H. Francisco Ruiz Paredes
    • 3
  1. 1.Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Osaka, Japan, e-mail: nhayashi@math.wani.osaka-u.ac.jpJP
  2. 2.Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, MéxicoMX
  3. 3.Programa de Graduados de Eléctrica, Instituto Tecnológico de Morelia, CP 58120, Morelia, Michoacán, MéxicoMX

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