April  1999, 5(2): 375-390. doi: 10.3934/dcds.1999.5.375

Uniqueness and long-time behavior for the conserved phase-field system with memory

1. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

2. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

3. 

Institut Elie Cartan, Université de Nancy 1, B.P. 239, 54506 Vandœuvre les Nancy Cedex, France

4. 

Department of Mathematics, Technion-IIT, Haifa 32000, Israel

Received  June 1998 Revised  November 1998 Published  January 1999

This paper is concerned with a conserved phase-field model with memory. We include memory by replacing the standard Fourier heat law with a constitutive assumption of Gurtin-Pipkin type, and the system is conservative in the sense that the initial mass of the order parameter as well as the energy are preserved during the evolution. A Cauchy-Neumann problem is investigated for this model which couples a Volterra integro-differential equation with fourth order dynamics for the phase field. A sharp uniqueness theorem is proven by demonstrating continuous dependence for a suitably weak formulation. With regard to the long-time behavior, the limit points of the trajectories are completely characterized.
Citation: Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375
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