Wide oscillation finite time blow up for solutions to nonlinear fourth-order differential equations
type: Item
instance of: scholarly article
title: Wide oscillation finite time blow up for solutions to nonlinear fourth-order differential equations
zbMATH Open document ID: 1278.34036
published in: Archive for Rational Mechanics and Analysis
publication date: 2013-03-27T00:00:00Z
review text: The paper studies in finite time oscillating blow up solutions to the fourth-order nonlinear ordinary differential equation \[ w''''(s)+k w''(s)+f(w(s))=0, \tag{1} \] where \(k\in\mathbb R\) and \(f\) is a locally Lipschitz function. ``The primary purpose of the paper is to contribute to a better understanding of the qualitative properties of solutions to (1) when the nonlinearity \(f\) satisfies \[ f\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb R), \quad f(t)t>0\text{ for }t\in\mathbb R\setminus \{0\}. \] Further assumptions on \(f\) are needed in the sequel, although the prototype nonlinearity investigated here is \[ f(t)=\alpha|t|^{q-1}t+|t|^{p-1}t,\quad p>q\geq 1,~ \alpha\geq 0. \] The second, and probably most ambitious, purpose of the paper is to connect the phenomena which hold for (1) with several classes of fourth-order partial differential equations.'' The authors also provide several open problems and numerical results which describe this oscillating blow up.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 6148933
zbMATH Keywords: blow up
zbMATH Keywords: oscillation
zbMATH Keywords: fourth-order differential equation
zbMATH Keywords: bridge model
author: Filippo Gazzola
author: Raffaella Pavani
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
full work available at URL: s00205-012-0569-5
OpenAlex ID: W2033032867
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DOI: 10.1007/S00205-012-0569-5
description: scientific article; zbMATH DE number 6148933