A concept of solution and numerical experiments for forward-backward diffusion equations
type: Item
instance of: scholarly article
title: A concept of solution and numerical experiments for forward-backward diffusion equations
zbMATH Open document ID: 1105.35007
author: Nicola Guglielmi
author: Giovanni Bellettini
author: Giorgio Fusco
published in: Discrete and Continuous Dynamical Systems
publication date: 2007-02-05T00:00:00Z
review text: The authors study the gradient flow associated with the functional \(F_{\phi} (u): = \frac12 \int_I \phi (u_x)\, dx\), where \(\phi\) is non convex, and with its singular perturbation \(F_{\phi}^x (u): = \frac12 \int_I (\varepsilon^2 (u_{xx})^2 + \phi (u_x))\,dx\). With the support of numerical simulations, various aspects of the global dynamics of solutions \(u^{\varepsilon}\) of the singularly perturbed equation \(u_t= -\varepsilon^2 u_{xxxx} +\frac12 \phi'' (u_x) u_{xx}\) for small values of \(\varepsilon > 0\) are discussed. Their analysis leads to a reinterpretation of the unperturbed equation \(u_{tt} =\frac12 (\phi' (u_x))_x\), and to a well defined notion of a solution. Examine the conjecture that this solution coincides with the limit of \(u^{\varepsilon}\) as \(\varepsilon \to 0^+\) is given.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 34E13
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 5121942
reviewed by: Qin Meng Zhao
MaRDI profile type: Publication
full work available at URL: dcds.2006.16.783
OpenAlex ID: W2065526717
DOI: 10.3934/DCDS.2006.16.783
description: scientific article; zbMATH DE number 5121942