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

¹ School of Economics and Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China.
² Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA.
³ School of Mathematics, Sichuan University, Chengdu 610064, China.

^** Corresponding author: zxzhang@amss.ac.cn

Received: 3 March 2018
Accepted: 23 April 2019

Abstract

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 space H^s(0, L) for any s > −2 with the initial data in H^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 space H^s(0, L) for any s < −2 in the sense that the corresponding solution map fails to be in C².