On the finite space blow up of the solutions of the Swift-Hohenberg equation

TL;DR

This paper investigates the conditions under which solutions to a class of fourth order differential equations, specifically the Swift-Hohenberg equation, blow up in finite space, providing insights into oscillation behaviors and nonexistence of certain wave profiles.

Contribution

It confirms a conjecture about finite space blow up for solutions of fourth order equations and explores implications for beam oscillation and traveling wave profiles.

Findings

Confirmed a conjecture on finite space blow up.

Linked blow up behavior to nonexistence of low-speed traveling waves.

Provided new conditions for solution blow up in the Swift-Hohenberg equation.

Abstract

The aim of this paper is to study the finite space blow up of the solutions for a class of fourth order differential equations. Our results answer a conjecture in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717--752, 2013] and they have implications on the nonexistence of beam oscillation given by traveling wave profile at low speed propagation.