Remarks on time map for quasilinear equations
type: Item
instance of: scholarly article
title: Remarks on time map for quasilinear equations
zbMATH Open document ID: 1221.34054
author: Tomasz Adamowicz
author: Philip Korman
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-02-09T00:00:00Z
review text: This paper deals with one-dimensional quasilinear equations \[ \varphi (u')' + f(u) = 0, \;u(0) = 0, \;u'(0) = r > 0 \] including the case of the \(p\)-Laplacian, where \(\varphi (s) = | s | ^{p-2}s, \;p > 1.\) As the authors explain using a ballistic analogy, the previous initial value problem can be interpreted as shooting from the ground level at an angle \(r > 0.\) Then, the time map \(T = T(r)\) gives the time when the projectile falls back to the ground. In the paper, two different generalizations of Schaaf's time formula [\textit{R. Schaaf}, Global solution branches of two point boundary value problems. Lecture Notes in Mathematics. 1458. Berlin etc.: Springer-Verlag (1990; Zbl 0780.34010)] are presented. The first is obtained by globally \(p\)-linearizing the equation. The second involves parametric integration of the equation. These two formulas can be used to prove uniqueness or multiplicity results for the corresponding Dirichlet boundary value problem and for the computation of bifurcation diagrams.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5848912
zbMATH Keywords: quasilinear equations
zbMATH Keywords: one-dimensional case
zbMATH Keywords: time map formula
zbMATH Keywords: \(p\)-Lapacian operator
zbMATH Keywords: multiplicity results
zbMATH Keywords: bifurcation diagrams
reviewed by: Antonio Cañada
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.11.010
OpenAlex ID: W2054128297
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DOI: 10.1016/J.JMAA.2010.11.010
description: scientific article; zbMATH DE number 5848912