Dissipativity of a conserved phase-field system with memory

Federico Mario Vegni

doi:10.3934/dcds.2003.9.949

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2003, Volume 9, Issue 4: 949-968. Doi: 10.3934/dcds.2003.9.949

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Dissipativity of a conserved phase-field system with memory

Federico Mario Vegni^1,

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

Received: June 30, 2001

Revised: September 30, 2002

Published: June 2003

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Abstract

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 $a$.

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Mathematics Subject Classification: 35G25, 45K05, 80A22.

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