Viscous scalar conservation law with nonlinear flux feedback and global attractors
type: Item
instance of: scholarly article
title: Viscous scalar conservation law with nonlinear flux feedback and global attractors
zbMATH Open document ID: 0919.35061
author: K. Appert
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-03-23T00:00:00Z
review text: The authors study the forced viscous scalar conservation law on \((0,1)\) \[ u_t-\nu u_{xx}+f(u)_x=F, \;u=u(x,t), \;x\in (0,1), \;t>0 \] with initial data \(u(x,0)=u_0(x)\) and nonlinear boundary feedback control \[ u_x(0,t)-g_0(u(0,t))=u_x(1,t)+g_1(u(1,t))=0 \] where \(g_0\), \(g_1\) are continuous nondecreasing functions with \(g_0(0)=g_1(0)=0\). On the base of nonlinear semigroup theory global existence and uniqueness of the solution are established for any bounded measurable initial and forced functions. Under appropriate growth conditions on \((f,g_0,g_1)\) existence of an absorbing set and a nonempty compact global attractor in \(L^\infty\)-topology are also proved.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 1236175
zbMATH Keywords: nonlinear boundary feedback
zbMATH Keywords: absorbing sets
zbMATH Keywords: nonlinear semigroup theory
reviewed by: Evgeniy Panov
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6016
OpenAlex ID: W2017115981
cites work: On the global dynamics of a controlled viscous Burgers' equation
cites work: Q4040820
cites work: Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces
cites work: Pointwise control of Burgers' equation -- a numerical approach
cites work: Global solutions of the time-dependent drift-diffusion semiconductor equations
cites work: Asymptotic behavior of the drift-diffusion semiconductor equations
cites work: Q3960389
cites work: Q4040640
cites work: A Dissipative Feedback Control Synthesis for Systems Arising in Fluid Dynamics
cites work: FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES
cites work: Q5562267
cites work: Q3247330
cites work: An application of maximal dissipative sets in control theory
cites work: Q4746129
cites work: Q3207428
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Dynamical systems and evolution equations. Theory and applications
cites work: Q5332803
DOI: 10.1006/JMAA.1998.6016
description: scientific article; zbMATH DE number 1236175