Blow-up profile for solutions of a fourth order nonlinear equation. (July 2015)

• Record Type: Journal Article
• Authors: D'Ambrosio, Lorenzo; Lessard, Jean-Philippe; Pugliese, Alessandro
• Is Part Of: Nonlinear analysis. Volume 121(2015)
• Page Start: 280
• Page End: 335
• Publication Date: 2015-07
• Subjects: Suspension bridges -- Fisher–Kolmogorov -- Swift–Hohenberg -- Blow-up profile -- Computer assisted proof; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.12.026
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: It is well known that the nontrivial solutions of the equation u ⁗ ( r ) + κ u ″ ( r ) + f ( u ( r ) ) = 0 blow up in finite time under suitable hypotheses on the initial data, κ and f . These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher–Kolmogorov equation or Swift–Hohenberg equation. In this paper we provide details of the blow-up profile. The key idea is to relate this blow-up profile to the existence of periodic solutions for an auxiliary equation.