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Uniqueness and \longtime\ \bhv
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for the conserved \phf\ system with memory
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\author
Pierluigi Colli\thinspace,\quad Gianni Gilardi
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\address
Dipartimento di Matematica ``F. Casorati''

Universit\`a di Pavia, Via Ferrata 1, 27100 Pavia, Italy

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\author
Philippe Lauren\c cot
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Institut Elie Cartan, Universit\'e de Nancy 1

B.P.~239, 54506 Vand\oe uvre les Nancy Cedex,
France

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Amy  Novick--Cohen
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\address
Department of Mathematics,

Technion--IIT, Haifa 32000, Israel

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\noindent \bf
Abstract.
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This paper is concerned with a conserved \phf\ 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 \longtime\
\bhv, the limit points of the trajectories are completely characterized.
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\footnote{}
{\hskip-\parindent\smallfont{\bfs AMS Subject Classification.} 35R99,
45K05, 80A22.}

\footnote{}
{\hskip-\parindent\smallfont{\bfs Key Words.} Phase--field equations,
memory effects, initial--boundary value problem, uniqueness, 
long-time \bhv.}

\footnote{}
{\hskip-\parindent\smallfont{\bfs Acknowledgment.} This work has been
partially supported by the Istituto di Analisi Numerica of CNR in Pavia.  
Part of this work was done while Philippe Lauren\c cot enjoyed the 
hospitality of the Weierstra\ss--Institut f\"ur Angewandte Analysis und 
Stochastik in Berlin. The research of Amy Novick--Cohen 
was partially supported by the National
Science Foundation through the Center for Nonlinear Analysis.
Also, A. Novick--Cohen is grateful for the hospitality she enjoyed 
at the Applied Math. Dept. of the State University of
Campinas~(UNICAMP) while completing part of this work.}

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