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Initial-boundary value problem for the Korteweg-de Vries-Burgers equation without smallness condition on the data
We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation¶¶ $ \left\{\kern-2pt \begin{array}{l} u_{t}+{\it u}_{x}-u_{{\it xx}}+u_{{\it xxx}}=0, t>0,x>0, \\u(x,0)=u_{0}(x),\ x>0,\ u(0,t) =0,\ t>0. \end{array} \right. \label{KdV} $ ¶¶We prove that if the initial data $ u_{0}\in \mathbf{H}^{0,\omega }\cap \mathbf{H}^{1,0} $ , where $ \omega\in(\frac{1}{2},\frac{3}{2}) $ , then there exists a unique solution $ u\in \mathbf{C} \ ([ 0,\infty),\mathbf{H}^{1,\omega}) $ of the initial-boundary value problem (\ref{KdV}). Moreover if the initial data are such that $ x^{1+\mu }u_{0}(x)\in \mathbf{L}^{1},$ $\mu =\omega -\frac{1}{2}, $ then there exists a constant A such that the solution has the following asymptotics¶¶ $ u(x,t)=\frac{A}{t}\frac{x}{2\sqrt{\pi t}}e^{-\frac{x}{4t}^{2}}+O\left( \min \left(\frac{x}{\sqrt{t}},1\right) t^{-1-\frac{\mu }{2}}\right) $ ¶¶for $ t\rightarrow \infty $ uniformly with respect to x > 0.
Journal of Evolution Equations – Springer Journals
Published: Aug 1, 2002
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