A non-homogeneous boundary value problem for the Kuramoto-Sivashinsky equation posed in a finite interval

This paper studies the initial boundary value problem (IBVP) for the dispersive Kuramoto-Sivashinsky equation posed in a finite interval (0,L) with non-homogeneous boundary conditions. It is shown that the IBVP is globally well-posed in the spaceH(s)(0,L) for anys> -2 with the initial data inH(s)(0,L) and the boundary value data belonging to some appropriate spaces. In addition, the IBVP is demonstrated to be ill-posed in the spaceH(s)(0,L) for anys< -2 in the sense that the corresponding solution map fails to be inC(2).