Lyapunov functionals for reaction–diffusion equations with memory

✍ Scribed by Stefania Gatti; Maurizio Grasselli; Vittorino Pata

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 115 KB
• Volume: 28
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.635

✦ Synopsis

We consider a reaction-di usion equation in which the usual di usion term also depends on the past history of the di usion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the ÿrst results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, deÿnes a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a di erent method based on the notion of trajectory attractors. In addition, those authors provide su cient 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.