Long time small solutions to nonlinear parabolic equations
type: Item
instance of: scholarly article
title: Long time small solutions to nonlinear parabolic equations
zbMATH Open document ID: 0733.35061
DOI: 10.1007/BF02387387
published in: Arkiv för Matematik
publication date: 1990-01-01T00:00:00Z
review text: The existence of a unique global classical solution to the Cauchy problem \[ u_ t=\Delta u+f(u,Du,D^ 2u,u_ t),\quad u(0)=u_ 0\text{ in } {\mathbb{R}}^ n\times {\mathbb{R}}_+ \] is obtained under the assumption that f is \(C^{1+r}\) for \(r>2/n\) and \(u_ 0\) is small in \(C^ 2\) and \(W^{1,2}\)-norms. The result follows from an a priori estimate for solutions to the linear heat equation by using the Banach fixed point theorem.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 4212610
zbMATH Keywords: existence
zbMATH Keywords: global classical solution
zbMATH Keywords: Cauchy problem
zbMATH Keywords: a priori estimate
zbMATH Keywords: Banach fixed point theorem
author: Zhi-Min Chen
MaRDI profile type: Publication
cites work: Q5547841
cites work: Long-time behavior of solutions to nonlinear evolution equations
cites work: Global existence of small solutions to a class of nonlinear evolution equations
cites work: Existence and non-existence of global solutions for a semilinear heat equation
cites work: Global existence for nonlinear parabolic equations
description: scientific article; zbMATH DE number 4212610