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

Abstract.

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation on half-line¶¶\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}^{+}}$.}¶¶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.