On a simple maximum principle technique applied to equations on the circle
type: Item
instance of: scholarly article
title: On a simple maximum principle technique applied to equations on the circle
zbMATH Open document ID: 1154.35012
author: Yu-Chu Lin
author: Dong-Ho Tsai
published in: Journal of Differential Equations
publication date: 2008-07-10T00:00:00Z
review text: The authors consider, on the unit circle \(S^1\), the general quasilinear equation of the form \[ \begin{aligned} \frac{\partial w}{\partial t}(x,t)&= A(w)\cdot w_{xx}+B(w)\cdot w_x+C(w), \\ w( x,0)&= w_0(x), \quad x\in S^1, \end{aligned} \tag{1} \] where \((x,t)\in S^1\times[0,T)\), \(T>0\), \(A(\theta), B(\theta), C(\theta):\mathbb{R} \rightarrow \mathbb{R}\) are given but arbitrary smooth functions, \(w_0(x) \) is a given smooth function on \(S^1\). In order for \((1)\) to be parabolic, the authors consider that for the initial data \( w_0(x)\) the parabolic condition \(A(w_0(x))\geq C>0\) for all \(x\in S^1\), is satisfied, where \(C\) is some positive constant. As a consequence the authors obtain the existence for a unique solution \(w(x,t) \) which is smooth and is defined on \( S^1\times[ 0,T) \) for some finite \(T>0.\) Supposing that \(w\) is bounded on \(S^1\times[0,T)\), the purpose of the paper is to show, using the maximum principle technique, that all higher derivatives (space and time) of \(w\) are also bounded on \( S^1\times[0,T).\) The authors give some specific examples and under some suitable assumptions they derive exponential decay of the derivatives of the solution.
reviewed by: Cristian Chifu
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 5299424
zbMATH Keywords: quasilinear equation
zbMATH Keywords: exponential decay of the derivatives
MaRDI profile type: Publication
full work available at URL: j.jde.2008.04.007
OpenAlex ID: W1996244769
cites work: Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations
cites work: Expansion of embedded curves with turning angle greater than \(-\pi\)
cites work: Convergence, asymptotic periodicity, and finite-point blow-up in one- dimensional semilinear heat equations
cites work: Geometric expansion of convex plane curves
cites work: The heat equation shrinking convex plane curves
cites work: Four-manifolds with positive curvature operator
cites work: A CERTAIN PROPERTY OF SOLUTIONS OF PARABOLIC EQUATIONS WITH MEASURABLE COEFFICIENTS
cites work: Q5548525
cites work: Geometric expansion of starshaped plane curves
cites work: Behavior of the gradient for solutions of parabolic equations on the circle
DOI: 10.1016/J.JDE.2008.04.007
description: scientific article; zbMATH DE number 5299424