Remarks on Third-Order ODEs Relevant to the Kuramoto–Sivashinsky Equation

We are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto–Sivashinsky equation. So-called steady solutions of the Kuramoto–Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited.