Two initial--boundary-value problems for the Korteweg--de~Vries equation in a half-strip with two boundary conditions and in a bounded rectangle are considered and results on local and global well-posedness of these problems are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a ``boundary potential" type for this linearized KdV equation.
"Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation." Differential Integral Equations 20 (6) 601 - 642, 2007.