Premium
Lyapunov functionals for reaction–diffusion equations with memory
Author(s) -
Gatti Stefania,
Grasselli Maurizio,
Pata Vittorino
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.635
Subject(s) - dissipative system , mathematics , attractor , reaction–diffusion system , lyapunov function , diffusion , diffusion equation , dynamical systems theory , trajectory , term (time) , mathematical analysis , boundary (topology) , nonlinear system , physics , economy , quantum mechanics , astronomy , economics , thermodynamics , service (business)
We consider a reaction‐diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to asuitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions. Copyright © 2005 John Wiley & Sons, Ltd.