A nonhomogeneous boundary-value problem for the 5-th Korteweg–de Vries equation posed on the half line

Abstract

We study the well-posedness of the 5-th Korteweg–de Vries (KdV) equation in the space $H^{2+s}\left(0,r\right)$ when the initial data is drawn from $H^{2+s}\left(0,r\right)$ and the boundary data $\left(h_1\left(t\right),h_2\left(t\right),\dots ,h_5\left(t\right)\right)$ lies in the product space $H^{s_1}\left(0,T\right),\dots ,H^{s_5}\left(0,T\right)$ for some appropriate indices $s_1,s_2,\dots ,s_5$ that depend on $s$. As we will see later, the natural choices of $s_1,s_2,\dots ,s_5$ are $s_1=\left(s+2\right)∕5$, $s_2=\left(s+1\right)∕5$, $s_3=s∕5$, $s_4=\left(s-1\right)∕5$, $s_5=\left(s-2\right)∕5$. We first use contraction mapping method to obtain the local solution of the IBVP, furthermore to get the global solution by the a prior estimate.