An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator
type: Item
instance of: scholarly article
title: An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator
zbMATH Open document ID: 1057.65103
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-16T00:00:00Z
review text: Motivated by the porous media equation, as well as by problems of thermoelasticity and viscoelasticity, the author presents an analysis of Rothe's method applied to the (variational form of the) degenerate parabolic problem with memory term, \[ \partial_t\beta(u(t))-\Delta u(t)- \int^t_0 a(t,s)\Delta u(s)\,ds= f(t)\quad\text{in }(0,T)\times \Omega, \] with bounded \(\Omega\subset \mathbb{R}^d\) and subject to initial and (homogeneous) boundary conditions. The function \(\beta\) is assumed to be Lipschitz continuous (with Lipschitz constant \(C\)) and such that \[ \beta(0)= 0,\quad \beta'\geq 0\quad\text{a.e. in }\mathbb{R},\quad |\beta(s)|\geq C_1| s|- C. \] The convergence analysis is carried out for problems with convolution kernels \(a(t, s)= a(t- s)\) that are either smooth or weakly singular and completely monotonic.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 74S05
Mathematics Subject Classification ID: 74C10
zbMATH DE Number: 2091481
zbMATH Keywords: degenerate parabolic integrodifferential equation
zbMATH Keywords: time discretization
zbMATH Keywords: Rothe's method
zbMATH Keywords: porous media equation
zbMATH Keywords: thermoelasticity
zbMATH Keywords: viscoelasticity
zbMATH Keywords: convergence
zbMATH Keywords: convolution kernels
author: M. Slodička
reviewed by: H. Brunner
MaRDI profile type: Publication
full work available at URL: j.cam.2003.12.007
OpenAlex ID: W1998249980
Wikidata QID: Q115359903
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DOI: 10.1016/J.CAM.2003.12.007
description: scientific article; zbMATH DE number 2091481