Dissipativity of a conserved phase-field system with memory

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July 2003, 9(4): 949-968. doi: 10.3934/dcds.2003.9.949

Dissipativity of a conserved phase-field system with memory

1.	Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti, 9 I-25133 Brescia, Italy

Received July 2001 Revised October 2002 Published April 2003

Abstract
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We analyze a conserved phase-field system, characterized by heat memory terms: memory kernels $a$ and $b$ account for relaxation effects in the energy constitutive equation and in the Gurtin-Pipkin heat conduction law, respectively. This model consists of a hyperbolic integrodifferential equation for the temperature $\theta$ coupled with a nonlinear fourth order evolution equation for the phase variable $\chi$. With appropriate initial and boundary data, we prove that the system can be interpreted as a process. We show that it possesses an absorbing set, under three -thermodynamically consistent- conditions on the memory kernel $a$.

Keywords: continuous dependence, materials with memory, existence and uniqueness, nonautonomous dynamical systems, uniform energy estimates, Conserved phase-ﬁeld models, uniform absorbing set..

Mathematics Subject Classification: 35G25, 45K05, 80A2.

Citation: Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949

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