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

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation on half-line¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \cases{ u_{t}+u_{x}^{2}-u_{xx}+u_{xxx}=0, $(x,t)\in {{\bf R}^{+}}\times {{\bf R} ^{+}}$,\cr u(x,0)=u_{0}(x),$ x\in {{\bf R}^{+}}$,\hspace*{8pc}(1)\cr u(0,t)=0,$ t\in {{\bf R}^{+}}$.} \end{document}¶¶We prove that if the initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ u_{0}\in {\bf X} $\end{document}, and the norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \Vert u_{0}\Vert _{{\bf X}} $\end{document} is sufficiently small, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\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}}$\end{document}<\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \infty \} $\end{document}, then there exists a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ u\in {\bf C}([0,\infty );{\bf H}^{1}) $\end{document} of the initial-boundary value problem (1), where Hk is the Sobolev space with norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \Vert \phi \Vert _{{\bf H}^{k}}=\Vert (1-\partial _{x}^{2})^{\frac{k}{2}}\phi \Vert _{{\bf L}^{2}}. $\end{document} We also find the large time asymptotics of the solutions under the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ x^{1+\delta }u\in {\bf L}^{1}\cap {\bf L}^{2} $\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \delta \in (0,1). $\end{document} More pricesely, we prove¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ 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), $\end{document}¶¶where A will be defined below in Theorem 2.