A note on the Kuramoto-Sivashinsky equation with discontinuity

Abstract

In this work we consider differential equations of the type $$\pm\, u^{(k)}=f(u),$$ and study the extinction profile of their solutions. Emphasis is placed on the special case $-u^{(4)}=\sign(u)$, which is related to the Kuramoto-Sivashinsky equation. In this case we describe in more detail the extinction phenomenon and prove a conjecture by Galaktionov and Svirshchevskii.

Keywords

• discontinuous differential equations
• periodic solutions
• finite time extinction
• extinction profile
• oscillations