Global solutions to semilinear parabolic systems for small data
type: Item
instance of: scholarly article
title: Global solutions to semilinear parabolic systems for small data
zbMATH Open document ID: 0669.35046
DOI: 10.1016/0022-0396(88)90078-2
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: We consider semilinear parabolic systems \(u_ t+Au+f(u)=g,\) \(u(0)=u_ 0\), -A being the generator of an analytic semigroup with spectrum \(\sigma\) (A) in the right half plane. For appropriate small data \((g,u_ 0)\) global solutions are obtained if 0 belongs to the resolvent set and local existence is proved if \(0\in \sigma (A)\). The inverse-function theorem is used in a class of functions which are Hölder continuous with respect to time t. The asymptotic behaviour of the solution is given too. As first application we prove the existence of global (respectively local) solutions to the Navier-Stokes equations in \(L_ p\)-spaces in bounded (respectively unbounded) domains. Secondly we consider the case where A is an elliptic operator of order 2m and f has bounded growth. Thirdly a quasilinear example where the nonlinearity involves derivatives of the same order as A is studied. Then an example of a nonautonomous nonlinearity is given and finally the convergence of solutions of the time-dependent system to solutions of the stationary (semilinear elliptic) one is derived for \(0\in \rho (A)\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4096009
zbMATH Keywords: semilinear
zbMATH Keywords: generator
zbMATH Keywords: analytic semigroup
zbMATH Keywords: small data
zbMATH Keywords: global solutions
zbMATH Keywords: local existence
zbMATH Keywords: Hölder continuous
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: order 2m
zbMATH Keywords: bounded growth
zbMATH Keywords: quasilinear
zbMATH Keywords: nonautonomous nonlinearity
zbMATH Keywords: convergence
zbMATH Keywords: stationary
author: Reinhard Racke
reviewed by: Reinhard Racke
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90078-2
OpenAlex ID: W2028728107
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