Asymptotic behaviour of solutions to semi-linear elliptic equations on the half-cylinder
type: Item
instance of: scholarly article
title: Asymptotic behaviour of solutions to semi-linear elliptic equations on the half-cylinder
zbMATH Open document ID: 0762.35012
DOI: 10.1007/BF00946237
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1992-09-27T00:00:00Z
review text: The authors consider a non-prismatic cylinder of semi-infinite length with variable cross-section and plane base, \(\Omega=\{x\in\mathbb{R}^ 3:(x_ 2,x_ 2)\in D(x_ 3)\), \(x_ 3\in\mathbb{R}+\}\), where the bounded, plane cross-section \(D(x_ 3)\) has smooth boundary, and area \(| D(x_ 3)|\). On \(\Omega\), \(u\) satisfies the semi-linear equation \((a_{ij}u,_ i),_ j=f(u)\), with \(a_{ij}\) symmetric and uniformly elliptic. Further, \[ \lim_{x_ 3\to\infty}\int^{x_ 3}_ 0A^{- \mu}(\nu)d\nu=\infty,\quad\mu=2p/(1+p), \] where \(A(x_ 3)=| D(x_ 3)|^{(p-1)/2p}a_ 1^{1/2}(x_ 3)\), \(p>1\), and \(a_ 1(x_ 3)=\max_{X_ \alpha\in D(x_ 3)}a_{33}(x_ \alpha,x_ 3)\). The boundary conditions are either Dirichlet or Neumann. The main result of the paper is the following. If \(f\) satisfies \(uf(u)\geq c| u|^{2p}\), \(x_ 3>0\), then a smooth solution either fails to exist globally, or else it must tend asymptotically to zero as \(x_ 3\to\infty\). Various extensions are also discussed, including alternate conditions on the nonlinearity, and elliptic systems.
reviewed by: James R. Kuttler
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 63807
zbMATH Keywords: homogeneous Dirichlet boundary conditions
zbMATH Keywords: different classes of nonlinearities
zbMATH Keywords: derivation of first order nonlinear differential inequality
zbMATH Keywords: Neumann problem
author: J. N. Flavin
author: R. J. Knops
author: L. E. Payne
MaRDI profile type: Publication
cites work: Phragmén-Lindelöf theorems for nonlinear elliptic equations
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cites work: Decay theorems for nonlinear Dirichlet problems in semi-infinite cylinders
cites work: Decay estimates for the constrained elastic cylinder of variable cross section
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cites work: Lower Bounds for Stekloff and Free Membrane Eigenvalues
cites work: On the Nonexistence of Entire Solutions to Nonlinear Second Order Elliptic Equations
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cites work: An optimal Poincaré inequality for convex domains
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description: scientific article; zbMATH DE number 63807