Oscillation of nonlinear parabolic equations with functional arguments
type: Item
instance of: scholarly article
title: Oscillation of nonlinear parabolic equations with functional arguments
zbMATH Open document ID: 0614.35048
author: Norio Yoshida
published in: Hiroshima Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Some results are established concerning the oscillation of solutions to the nonlinear parabolic equations \[ u_ t=a(t)\Delta u\pm q(x,t)f(u(x,\sigma (t))),\quad (x,t)\in \Omega \times {\mathbb{R}}_+, \] where \(\Delta\) is the Laplacian in \({\mathbb{R}}^ n\) and \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\). The functional argument \(\sigma\) (t) is required to be continuous and satisfy \(\lim_{t\to \infty}\sigma (t)=\infty\). The solution u is oscillatory provided it has a zero in \(\Omega\times [\tau,\infty)\) for every \(\tau >0\). Two types of boundary conditions are imposed, either \(u=0\) or \(\partial u/\partial \nu +\mu u=0\) on \(\partial \Omega \times {\mathbb{R}}_+\). The method of proof is to reduce the problem to a one-dimensional oscillation problem for ordinary differential equations or differential inequalities involving the first eigenfunction of the problem \(\Delta v+\lambda v=0\) in \(\Omega\), \(v=0\) on \(\partial \Omega\). A number of examples are given to illustrate the results.
reviewed by: Glenn Webb
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 3994120
zbMATH Keywords: oscillation
zbMATH Keywords: functional argument
zbMATH Keywords: boundary conditions
zbMATH Keywords: differential inequalities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3994120