
\magnification=\magstep1
\vsize=23truecm
\hsize=16truecm

\hoffset=-0.075truecm
\voffset=-0.75truecm

\parindent=1truecm
\nopagenumbers

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       running title

\headline={%
  \ifnum\pageno>1
    \headfont
    \ifodd\pageno
      \hfil\headtitolo\hfil\llap{\folio}%
    \else
      \rlap{\folio}\hfil\headnomi\hfil
    \fi
  \else
    \hfil
  \fi}

\def\headtitolo{The conserved \phf\ system with memory}

\def\headnomi{Colli\enskip--\enskip Gilardi\enskip--\enskip
  Lauren\c cot\enskip--\enskip Novick--Cohen}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
%       fonts
%
\font\bigfont=cmbx10 at 17.28 truept    % title
\font\headfont=cmcsc10 at 10.00 truept  % headline
\font\author=cmcsc10                    % we
\font\address=cmsl10 at 10.95 truept    % our address
\font\email=cmtt10 at 10.95truept       % our e-mail
\font\auts=cmcsc10 at 10.95 truept      % authors in references
\font\sls=cmsl10 at 10.95 truept        % titles in references
\font\bfs=cmbx10 at 10.95 truept        % volumes in references
\font\smallfont=cmr10 at 10.95 truept   % in our e-mail and in references

\let\fontparag\bf
\let\fontthm\bf

\font\fontquadre=cmtt10 scaled \magstephalf % for better `[' and `]'
               % Warning: if you don't have these fonts, trubles
               % arise. Then look for the word `['

\font\stt=cmtt10 at 8truept             % marginstyle

\font\tengr=msbm10      % fonts for sets of numbers
\font\sevengr=msbm7     % Warning: if you don't have these fonts, trubles
\font\fivegr=msbm5      % arise. Then look for \erre below

\font\truetenrm=cmr10 at 10.95 truept
\font\trueteni=cmmi10 at 10.95 truept
\font\trueseveni=cmmi7 at 7 truept

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
%                             M A C R O
%
%            This section is split into several subsections
%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%                     This is a Warning subsection

%       Some important control sequences have been redefined here.
%       They are \cases \eqno and \eqalignno.  Moreover, the new
%       control sequence \Cases has been defined.  Here are the
%       rules and the motivations.
%
%       The control sequences \cases and \Cases have to be used exactly as
%       the usual \cases of TeX.  Both produce an automatic \smallskip
%       between the lines and \Cases gives the display style to the math
%       part, in addition.
%
%       The control sequence \eqno can be followed either by a given text
%       as usual or by a new control sequence, e.g., \blabla.  In the
%       latter case, \blabla is automatically defined as the right number
%       of the formula, e.g., (5.7) if the previous one was (5.6).
%       Moreover the name \blabla of this formula appears in a margin
%       note (look for \definitivotrue to get rid of the notes).
%       If instead the argument after \eqno has already a meaning, a
%       message appears on the display.
%
%       A similar thing happens for \eqalignno which, however, forbids
%       some structures that TeX allows.  You must always use & twice
%       in each line (while TeX allows the number of &'s to be 0,1,2).
%       Moreover, if the second & is followed by a new control sequence,
%       don't write any blank space in between and the control sequence
%       becomes a number (like (5.7) as above) in the sequel.
%
%       Examples of good lines are the following ones
%
%         1st part of math formula (possibly empty)
%       & 2nd part of math formula (possibly empty)
%       & some text, e.g. (*)
%       \cr
%
%         1st part of math formula (possibly empty)
%       & 2nd part of math formula (possibly empty)
%       &\anynewcontrolsequence
%       \cr
%
%       while the line
%
%         1st part of math formula (possibly empty)
%       & 2nd part of math formula (possibly empty)
%       & \newname
%       \cr
%
%       is bad because of the blank space between & and \newname
%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       This subsection defines automatucally \aaa \bbb ... \ZZZ
%       as {\bf a} {\bf b} ... {\bf Z} respectively.
%       Similarly, \calA ... \calZ are defined as {\cal A} ... {\cal Z}

\def\bold#1{%
  \ifx#1\finecostruz
    \let\next\relax
  \else
    \let\next\bold
    \expandafter\def\csname#1#1#1\endcsname{{\bf #1}}
  \fi
  \next}
\def\finecostruz{\finecostruz}

\bold qwertyuiopasdfghjklzxcvbnmQWERTYUIOPASDFGHJKLZXCVBNM\finecostruz

\def\multical#1{%
  \ifx#1\finecostruz
    \let\next\relax
  \else
    \let\next\multical
    \expandafter\def\csname cal#1\endcsname{{\cal #1}}
  \fi
  \next}

\multical QWERTYUIOPASDFGHJKLZXCVBNM\finecostruz

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       greek letters

\let\eps\varepsilon
\let\ffi\varphi
\let\theta\vartheta

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\def\chi{{\setbox0                      % baseline
  \hbox{\mathsurround0pt $\TeXchi$}%    %
  \hbox{\raise\dp0 \copy 0 }}}          %

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       sets of numbers

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\def\grass{\fam\grfam\tengr}
\textfont\grfam=\tengr
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\scriptscriptfont\grfam=\fivegr

\def\enne{{\grass N}}     % Warning: replace these definitions with the
\def\erre{{\grass R}}     % ones given below if font troubles arise

%\def\erre{{\rm I\!R}}
%\def\enne{{\rm I\!N}}

%       better `[' and `]'

%       Warning: if troubles arise, comment with % from the first line
%       containing \catcode up to the line containing \def] (i.e., save
%       the definition \def[#1]{...})
%
%       use \[...] in place of [...] in a _part or to get the automatic
%       (usually bigger) size of [ and ] (don't use \left[...\right])

\let\apertaquadra[
\let\chiusaquadra]

\catcode`\[=13
\catcode`\]=13

\def[{{\fontquadre\lower.3pt\hbox{\apertaquadra}}}
\def]{{\fontquadre\lower.3pt\hbox{\chiusaquadra}}}

\def\[#1]{\left\apertaquadra#1\right\chiusaquadra}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       for the sake of convenience

\def\cases#1{\left\{\,\vcenter{\normalbaselines\mathsurround0pt
  \everycr{\noalign{\smallskip}}%
    \halign{$##\hfil$&\quad##\hfil\crcr#1\crcr}}\right.}
\def\Casesnograffa#1{\vcenter{\normalbaselines\mathsurround0pt
  \everycr{\noalign{\smallskip}}%
    \halign{$\displaystyle{{}##}\hfil$&\quad##\hfil\crcr#1\crcr}}}
\def\Cases#1{\left\{\,\Casesnograffa{#1}\right.}

\def\tonde#1{\left(#1\right)}
\def\graffe#1{\left\{#1\right\}}
\def\modulo#1{\left\vert#1\right\vert}
\def\norma#1{\left\Vert#1\right\Vert}

\def\normaV#1{\norma{#1}_V}
\def\normaH#1{\norma{#1}_H}
\def\normaW#1{\norma{#1}_W}
\def\normaVp#1{\norma{#1}_{V'}}
\def\normaWp#1{\norma{#1}_{W'}}
\def\normaLqO#1{\norma{#1}_{L^4(\Omega)}}
\def\normaLqQt#1{\norma{#1}_{L^4(\Omega\times(0,t))}}
\def\normaLqQT#1{\norma{#1}_{L^4(\Omega\times(0,T))}}

\def\<#1{%
  \def\arg{#1}%
  \ifx\arg\minore
    \let\next\doppiadualita
  \else
    \def\next{\dualita#1}%
  \fi
  \next}
\def\minore{<}
\def\dualita#1>{\langle#1\rangle}
\def\doppiadualita#1>>{\langle\neghskip\langle#1\rangle\neghskip\rangle}
\def\neghskip{\hskip-2pt}

\def\cpunto{{}\cdot{}}

\def\intO{\int_\Omega}
\def\intQinfty{\int_0^\infty \mskip-6mu \intO}
\def\intQT{\int_0^T \mskip-5mu \intO}
\def\intQt{\int_0^t \mskip-3mu \intO}
\def\intQtn{\int_0^{t_n+t} \mskip-5mu \intO}

\def\unmezzo{{1\over2}}

\let\el\ell
\def\ell{\el\,}

\let\vuoto\emptyset
\let\meno\setminus
\let\incluso\subseteq
\let\weak\rightharpoonup
\def\weakstar{\mathrel{\mathop\weak\limits^*}}
\def\seto{\mathrel{\scriptstyle\searrow}}
\def\esiste{\exists\,}
\def\perogni{\forall\,}

\def\_{\discretionary{-}{}{--}}
\def\pp #1.{\hbox{#1.}}

\def\pw{piece\_wise}
\def\lsc{lower\_semicontinuous}
\def\lscty{lower\_semicontinuity}
\def\wk{well\_known}
\def\sc{so\_called}
\def\longtime{long\_time}
\def\Longtime{Long\_time}
\def\phf{phase\_field}

\def\lhs{left hand side}
\def\rhs{right hand side}
\def\nbh{neighbourhood}
\def\sfw{straightforward}
\def\bhv{behavior}

\def\Lip{Lip\-schitz}
\def\omegalimit{$\omega-$\kern-\mathsurround limit}

\def\and{\quad\hbox{and}\quad}
\def\aand{\qquad\hbox{and}\qquad}

\let\TeXhat\widehat
\let\TeXtilde\widetilde
\def\hat#1{\mathstrut\smash{\TeXhat{#1}}}
\def\tilde#1{\mathstrut\smash{\TeXtilde{#1}}}

\def\grad{\mathop{\rm grad}\nolimits}
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\def\supp{\mathop{\rm supp}\nolimits}
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\def\partialn{\partial_\nu}
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\def\toinfty{\mathrel{\scriptstyle\nearrow}+\infty}
\def\ntoinfty{n\mathrel{\scriptstyle\nearrow}\infty}

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  {C^{#1}([0,#3];#2)}{C^{#1}([0,#3];#2)}{C^{#1}(\[0,#3];#2)}{}}
\def\C#1#2{\preC{#1}{#2}T}
\def\Ct#1#2{\preC{#1}{#2}t}
\def\Cs#1#2{\preC{#1}{#2}s}
\def\L#1#2{L^{#1}(0,T;#2)}
\def\Lt#1#2{L^{#1}(0,t;#2)}
\def\Ls#1#2{L^{#1}(0,s;#2)}
\def\LL#1#2{L^{#1}(0,\infty;#2)}
\def\W#1#2{W^{#1}(0,T;#2)}
\def\H#1#2{H^{#1}(0,T;#2)}

\def\ez{{\eta_0}}
\def\thetaz{{\theta_0}}
\def\chiz{{\chi_0}}
\def\chizq{{\chi_0^2}}
\def\chizc{{\chi_0^3}}
\def\omegaz{\omega(\thetaz,\chiz)}
\def\muz{\mu_0}

\def\diffchic{\chi_1^3-\chi_2^3}
\def\sqerror{\normaVp\thetaz^2+\normaVp\chiz^2+\norma g_{\L1{V'}}^2}

\def\thetai{{\theta_{\!\infty}}}
\def\chii{{\chi_\infty}}
\def\chiiq{{\chi_\infty^{\,2}}}
\def\chiic{{\chi_\infty^{\,3}}}
\def\wi{{w_\infty}}

\def\thetan{{\theta_n}}
\def\chin{{\chi_n}}
\def\chinc{{\chi_n^3}}
\def\ffin{{\ffi_n}}
\def\psin{{\psi_n}}
\def\xin{{\xi_n}}
\def\wn{{w_n}}

\def\thetab{{\overline\theta}}
\def\psib{{\overline\psi}}
\def\ffib{{\overline\ffi}}
\def\xib{{\overline\xi}}
\def\zetab{{\overline\zeta}}
\def\wb{{\overline w}}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       Warning: don't modify up to the words   End of Warning

\newcount\contaparag
\newcount\contathm
\newcount\contaeq
\newcount\contaref

\newif\ifdefinitivo

%       sections

\def\parag#1#2 \par{\spaziosezione
  \contaeq=0
  \contathm=0
  \global\advance\contaparag 1
  \xdef#1{\the\contaparag}
  \wlog{\string#1\space = #1}           % promemoria nel file.log
  \line{%
    \ifdefinitivo
    \else
      \llap{{\stt\string#1}}%
    \fi
    \hfil\fontparag\the\contaparag.\enskip #2\hfil}
  \nobreak\medskip\noindent\rm}

\def\spaziosezione{\removelastskip\goodbreak\vskip1truecm}

\def\references{\spaziosezione
  \mathsurround 0pt
  \frenchspacing
  \tolerance 1000
  \centerline{\fontparag References}
  \nobreak\smallskip
  \expandafter\cita\the\listaref}

%       theorems and so on

\def\sect #1. {\removelastskip\bigskip\noindent
  \ifdefinitivo
  \else
    \llap{{\stt\etichetta\ }}%
  \fi
  {\fontthm #1.}\quad}

\def\thm#1{\memo#1\sect Theorem #1. \sl}
\def\prop#1{\memo#1\sect Proposition #1. \sl}
\def\deff#1{\memo#1\sect Definition #1. \sl}
\def\lem#1{\memo#1\sect Lemma #1. \sl}
\def\cor#1{\memo#1\sect Corollary #1. \sl}
\def\rem#1{\memo#1\sect Remark #1. \rm}

\def\memo#1{\global\advance\contathm 1
  \xdef#1{\the\contaparag.\the\contathm}\def\etichetta{\string#1}%
  \wlog{\string#1\space = #1}}          % promemoria nel file.log

\newif\iffinemmode

\def\fine{%
  \ifmmode
    \global\finemmodetrue
  \else
    \global\finemmodefalse \medskip\rm
  \fi}

\def\Fine{\kern4pt \vrule height4pt depth0pt width4pt \fine}

\def\proof{\stileproof{Proof}}
\def\proofof#1. {\stileproof{Proof of #1}}

\def\stileproof#1{%
  \iffinemmode
  \else
    \removelastskip
  \fi
  \medskip\noindent{\bf #1.}\quad\rm}

%       formula numbers (new \eqno and \eqalignno)

\let\TeXeqno\eqno
\let\TeXeqalignno\eqalignno

\def\eqno#1{\global\advance\contaeq 1
  \ifx#1\undefined
  \xdef#1{\bgroup\noexpand\rm
    (\the\contaparag.\the\contaeq)\egroup\noexpand\adestra\noexpand#1}%
  \wlog{\string#1\space = (\the\contaparag.\the\contaeq)}%
  \else
    \def\gianota{#1}%
    \ifx\gianota\space
    \else
      \messaggiogianota
    \fi
  \fi
  \ifdefinitivo
    \def\adestra##1{}%
  \else
    \def\adestra##1{\rlap{\stt\string##1}}%
  \fi
  \TeXeqno#1}

\def\messaggiogianota{\wvd{}\wvd{ONE HAS ALREADY GIVEN A MEANING TO
  \expandafter\string\gianota\space }\wvd{}}

\def\wvd#1{\immediate\write 16 {#1}}

\let\TeXeqalignno\eqalignno

\def\eqalignno#1{\defs#1\finedefs{#1}}

\def\defs#1&#2&#3\cr#4{%
  \def\arg{#3}%
  \ifx#3\undefined
    \global\advance\contaeq 1
    \xdef#3{\bgroup\noexpand\rm
      (\the\contaparag.\the\contaeq)\egroup\noexpand\adestra\noexpand#3}%
    \wlog{\string#3\space = (\the\contaparag.\the\contaeq)}%
  \else
    \def\gianota{#3}%
    \ifx\gianota\space
    \else
      \messaggiogianota
    \fi
  \fi
  \ifx#4\finedefs
    \let\next\finedefs
  \else
    \def\next{\defs#4}%
  \fi
  \next}

\def\finedefs#1{%
  \ifdefinitivo
    \def\adestra##1{}%
  \else
    \def\adestra##1{\rlap{\stt\string##1}}%
  \fi
  \TeXeqalignno{#1}}

\def\adestra#1{}

\def\accorpa #1#2{%
  {%
   \rm\expandafter\expandafter\expandafter\ccrp\expandafter #1#2}}

\def\ccrp\bgroup\rm(#1.#2)\egroup
  \adestra#3\bgroup\rm(#4.#5)\egroup\adestra#6{(#1.#2--#5)}

\def\Def#1#2{\def#1{#2}\bgroup
  \def\rm{}\wlog{}\wlog{*** \string#1\space= #2}\wlog{}\egroup}

\catcode`\@=11
\def\displaylines#1{\displ@y
  \halign{\hbox to\displaywidth{$\@lign\hfil\displaystyle{}##\hfil$}\crcr
    #1\crcr}}
\catcode`\@=12

%       for references

\newtoks\listaref

\def\numera#1{\def\argomento{#1}%
  \ifx\argomento\fineelenco
    \let\next\concludinumerazione
  \else
    \def\next{\continuanumerazione#1}%
  \fi
  \next}

\def\fineelenco{\fineelenco}
\def\concludinumerazione{\listaref=\expandafter{\the\listaref\fineelenco}}

\def\continuanumerazione#1#2\par{\advance\contaref 1
  \edef#1{\the\contaref}%
  \wlog{\string#1\space = \string[#1\string]}%
  \listaref=\expandafter{\the\listaref#1#2\par}
  \numera}

\def\cita#1{\def\argomento{#1}%
  \ifx\argomento\fineelenco
    \let\next\relax
  \else
    \def\next{\continuacitazione#1}%
  \fi
  \next}

\def\continuacitazione#1#2: #3\par{\smallskip
  \item{%
    \ifdefinitivo
    \else
      \llap{{\stt\string#1}}%
    \fi
    [#1]}\auts#2: \smallfont #3\cita}

%                           End of Warning

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
%                            Very beginning
%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
%       Warning: first \definitivotrue or not (see the line below)
%                then the section of the references (next Warning)
%                then the body of the paper
%                then just the control sequence \references
%                finally, \bye
%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

\definitivotrue  % If you cancel the comment in front of \definitivotrue
                  % all margin notes will disappear

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       Warning: this section gives a number to each reference
%                and MUST be written exactly HERE

\numera
%
%       Warning: NO blank line between \numera and the first reference.
%
%                Moreover, each reference must have the following
%                structure, as done below:  new control sequence,
%                author(s), colon (:), then whatever you want, blank line
%                (otherwise troubles arise).
%
%                Finally, the list MUST end with \fineelenco
\ABarbu
S. Aizicovici and V. Barbu:
{\sls Existence and asymptotic results for a system of integro\_partial
differential equations,}
NoDEA Nonlinear Differential Equations Appl.,
{\bfs 3}
(1996),
\pp 1--18.

\Barbu
V. Barbu:
``Nonlinear semigroups and differential equations in Banach spaces'',
Noord\-hoff,
Leyden,
1976.

\Brezis
H. Br\'ezis:
``Op\'erateurs maximaux monotones et semi\_groupes de contractions
dans les espaces de Hilbert'',
North\_Holland Math. Stud.,
{\bfs 5},
North\_Holland,
Amsterdam,
1973.

\BHN
D. Brochet, D. Hilhorst, and A. Novick\_Cohen:
{\sls Maximal attractor and inertial sets for a conserved phase field
model,}
Adv. Differential Equations,
{\bfs 1}
(1996),
\pp 547--578.

\CLuno
P. Colli and Ph. Lauren\c cot:
{\sls Existence and stabilization of
solutions to the phase\_field model with memory,}
J. Integral Equations Appl.,
to appear.

\CLdue
P. Colli and Ph. Lauren\c cot:
{\sls Uniqueness of weak solutions
to the phase\_field model with memory,}
 J.  Math.  Sci.  Univ.  Tokyo,
to appear.

\GriNov
M. Grinfeld and A. Novick\_Cohen:
{\sls Counting stationary solutions of the Cahn\_Hilliard equation by
transversality arguments,}
Proc.  Roy.  Soc.  Edinburgh Sect.~A,
{\bfs 125}
(1995),
\pp 351--370.

\Haraux
A. Haraux:
``Syst\`emes dynamiques dissipatifs et applications'',
RMA Res. Notes Appl. Math.,
{\bf 17},
Masson,
Paris,
1991.

\KenNie
N. Kenmochi and M. Niezg\'odka:
{\sls Nonlinear system for non\_isothermal diffusive phase
separation,}
J. Math. Anal. Appl.,
{\bfs 188}
(1994),
\pp 651--679.

\KNP
N. Kenmochi, M. Niezg\'odka, and I. Pawlow:
{\sls Subdifferential operator approach to the Cahn\_Hilliard equation
with constraint,}
J.  Differential Equations,
{\bfs 117}
(1995),
\pp 320--356.

\Lions
J.L. Lions:
``Quelques M\'ethodes de R\'esolution des Probl\`emes aux Limites Non
Lin\'eai\-res'',
Dunod Gauthier--Villars,
Paris,
1969.

\NST
B. Nicolaenko, B. Scheurer, and R. Temam:
{\sls Some global dynamical properties of a class of pattern formation
equations,}
Comm. Partial Differential Equations,
{\bfs 14}
(1989),
\pp 245--297.

\NC
A. Novick\_Cohen:
{\sls Conserved phase\_field equations with memory,}
in ``Curvature flows and related topics'',
A.~Damlamian, J.~Spruck, and A.~Visintin eds.,
pp. 179--197,
GA\-KU\-TO Internat.  Ser.  Math.  Sci.  Appl.,
{\bfs 5},
Gakk\= otosho,
Tokyo,
1995.

\Pruss
J. Pr\"uss:
``Evolutionary Integral Equations and
Applications'',
Monogr. Math.,
{\bfs 87},
Birk\-h\"auser,
Basel,
1993.

\fineelenco

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

%       Title, authors, addresses

\bgroup                                 % begin center
\leftskip 0pt plus1fil
\rightskip\leftskip
\parindent 0pt
\parfillskip 0pt

\bigfont
Uniqueness and \longtime\ \bhv
\vskip0.3truecm

for the conserved \phf\ system with memory
\vskip1.3truecm
\rm

\author
Pierluigi Colli\thinspace,\quad Gianni Gilardi
\vskip0.1truecm

\address
Dipartimento di Matematica ``F. Casorati''

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

\vskip0.5truecm

\author
Philippe Lauren\c cot
\vskip0.1truecm

\address
Institut Elie Cartan, Universit\'e de Nancy 1

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

\vskip0.5truecm\noindent

\author
Amy  Novick--Cohen
\vskip0.1truecm

\address
Department of Mathematics,

Technion--IIT, Haifa 32000, Israel

\egroup                                 % end center

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\vskip2truecm
\noindent \bf
Abstract.
\rm
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.
\bigskip

\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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\parag\Intro
Introduction

The initial--boundary value problem in which we are interested is the
following
$$
  \eqalignno{
  (\theta + \ell \chi)_t - \Delta (k*\theta) = g
  & \qquad \hbox{in \ } \Omega \times (0,+\infty)
  &\Iuno
  \cr
  \chi_t - \Delta ( - \Delta \chi + \chi^3 - \chi - \ell \theta ) = 0
  & \qquad \hbox{in \ } \Omega \times (0,+\infty)
  &\Idue
  \cr
  \partialn {(k*\theta)}
  = \partialn \chi
  = \partialn {( - \Delta \chi + \chi^3 - \chi - \ell \theta )} = 0
  & \qquad \hbox{on \ } \Gamma \times (0,+\infty)
  &\Itre
  \cr
  \theta (0) = \thetaz, \qquad \chi(0) = \chiz
  & \qquad \hbox{in \ } \Omega
  &\Iquattro
  \cr
  }
$$
\Def\Ipbl{\accorpa\Iuno\Iquattro}%
where $\Omega$ is a bounded connected domain in $\erre^N$, $N=1,2,3$, with
a smooth boundary~$\Gamma$, $\partialn$~denotes the outward normal partial
derivative on~$\Gamma$, $\el$~is a positive constant representing the
latent heat, $k\in L^1(0,\infty)$ is a prescribed kernel, and we define
$$
  (k*\theta) (x,t)
  = \int_0^t k(t-s) \, \theta(x,s) \, ds,
  \qquad (x,t) \in \Omega \times (0,+\infty).
$$
Moreover, $g\in L^1(0,\infty;L^2(\Omega))$ is a known function depending
on the past history of~$\theta$ (up~to $t=0)$ and (possibly) on some heat
source, and $\thetaz,\chiz:\Omega\to\erre$ represent the initial data for
the (relative) temperature $\theta$ and the order parameter~$\chi$.

Here we study the \phf\ model \Ipbl\ under the physically motivated
assumption that $k$ is of positive type, that~is,
$$
  \int_0^T (k*v)(t) \, v(t) \, dt \geq 0
  \qquad \perogni v \in L^2(0,T), \quad \perogni T \in (0,+\infty).
  \eqno\Icinque
$$
If $\thetaz$ and $\chiz$ belong to $L^2(\Omega)$, Novick--Cohen derives in
[\NC] the existence of weak solutions to \Ipbl\ and proves a regularizing
effect for the solution component~$\chi$, ensuring that $\chi(t)\in
H^1(\Omega)$ for a.e.~$t>0$.  Moreover, in [\NC] uniqueness is shown when
either the latent heat $\el$ is sufficiently small or the kernel $k$ is
non\_negative, non\_increasing, and convex, with $k(0+)<+\infty$.

With the aim of focusing on the investigation of the asymptotic \bhv\ of
\Ipbl\ as $t$ goes to~$+\infty$, we are not concerned with the precise
regularity of the initial data $\thetaz$ and~$\chiz$, and thus we take
$\chiz\in H^1(\Omega)$ in addition, in order to guarantee that the
associated Liapunov functional is uniformly bounded on the entire
interval~$[0,+\infty)$.  Within this framework, we are able to show a sharp
uniqueness result, which holds for any $\el>0$ and for any kernel $k$ which
belongs to $L^1(0,T)$ for all $T\in(0,+\infty)$ and which
satisfies~\Icinque.  This is accomplished by establishing a continuous
dependence estimate involving weaker norms for both data and solutions
which enables the concept of solution to \Ipbl\ to be extended to a weaker
formulation as~well.

Now, we are going to develop three sections.  In the next one, we give an
explanation of our notation and the statement of our main results.  In
Section~3, we show the \Lip\ continuous dependence property discussed above
and obtain uniqueness.  Afterwards, Section~4 is devoted to the
\longtime\ \bhv\ of the system.  Following an argument devised in [\CLuno]
for the nonconserved \phf\ model with memory, we can completely describe
the structure of any cluster point $(\thetai,\chii)$ in the \omegalimit\
set as follows.  The function $\thetai$ is constant and takes the value
$$
  \thetai
  = {1 \over |\Omega|}
  \tonde{
    \intO \thetaz + \intQinfty g
  }
$$
while $\chii$ is a solution (in a suitable sense) of the steady state
equation for~\Idue.  Note that the uniqueness of~$\thetai$, which results
from mass and energy conservation, causes the function
$\theta(t)$ to converge to $\thetai$ as $t\toinfty$.

We point out that similar results on the asymptotic \bhv\ as $t$ goes
to~$+\infty$ have been developed for related problems.  For the reader's
convenience, we cite [\ABarbu,~\CLuno] for standard \phf\ models with
memory and [\KenNie,~\BHN] for the conserved \phf\ equations without memory
(indeed, in~[\BHN] the authors also study the structure of global
attractors).

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

\parag\Main
Main results

We set for convenience
$$
  H = L^2(\Omega), \qquad
  V = H^1(\Omega), \aand
  W = \graffe{v \in H^2(\Omega): \quad \partialn v = 0}.
$$
Let us identify $H$ with its dual space~$H'$, so that
$$
  W \subset V \subset H \subset V' \subset W'
  \eqno\inclusioni
$$
with dense and compact injections.  We denote by $(\cpunto,\cpunto)$ the
inner product in~$H$, by $\<\cpunto,\cpunto>$ the duality pairing between
$V'$ and~$V$, and by $\<<\cpunto,\cpunto>>$ the duality pairing between
$W'$ and~$W$.  Also, let $\calW$, $\calV$, $\calH$, $\calV'$, $\calW'$ be
the subspaces of $W$, $V$, $H$, $V'$, $W'$, respectively, of elements $v$
satisfying the null average condition, that~is,
$$
  \<< v , 1 >> = 0 \qquad
  (\hbox{note that }\ 1\in W).
$$
In particular, injections similar to~\inclusioni\ hold but, e.g., $\calV'$
is not identified here with the dual space of~$\calV$.

Now, we define the operator $\calN:\calH\to\calW$ that maps $v\in\calH$
into the unique function $\calN v\in H^2(\Omega)$ which solves
$$
  - \Delta (\calN v) = v
  \quad \hbox{a.e.~in } \Omega, \qquad
  \partialn (\calN v) = 0
  \quad \hbox{a.e.~on } \Gamma, \qquad
  \intO \calN v = 0.
  \eqno\dueuno
$$
This operator is an isomorphism and can be extended to a new operator
(still denoted by~$\calN)$ from $\calV'$ to $\calV$ by means of the
variational equality (here $v\in\calV')$
$$
  \calN v \in \calV, \qquad
  \intO \nabla (\calN v) \cdot \nabla z = \< v , z >
  \quad \perogni z \in \calV
  \eqno\duedue
$$
and further generalized to $v\in\calW'$ by transposition, as follows
$$
  \calN v \in \calH, \qquad
  - \intO (\calN v) \, \Delta z = \<< v , z >>
  \quad \perogni z \in \calW.
  \eqno\duetre
$$
Note that $\calN$ is also an isomorphism from $\calV'$ onto~$\calV$.  It
follows that, for $v\in\calV'$, the~norm
$$
  \tonde{
    \intO |\nabla (\calN v)|^2
  }^{\! 1/2}
  = \< v , \calN v >^{1/2}
  \qquad \hbox{is equivalent to} \quad \normaVp v
  \eqno\duequattro
$$
and we use this norm instead of $\normaVp v$ whenever it is more
convenient.  Similarly, $\calN$~is an isomorphism from $\calW'$ onto
$\calH$ and the~norm
$$
  \tonde{
    \intO |\calN v|^2
  }^{\! 1/2}
  \qquad \hbox{is equivalent to} \quad \normaWp v
  \eqno\duecinque
$$
for $v\in\calW'$.  Let us use the same notation $\normaH\cpunto$ both for
the norm in~$H=L^2(\Omega)$ and for that in~$H^N=L^2(\Omega,\erre^N)$.

Concerning the data, we recall that $\el$ is a positive constant and we
assume~that
$$
  k \in L^1(0,\infty), \qquad
    \int_0^\infty k(s) \, ds \not= 0
  \eqno\duesei
$$
and $k$ is a kernel of positive type, that is,
$$
  \int_0^T
    \bigl(
      v(t) , (k*v)(t)
    \bigr) \,dt
    \geq 0
    \qquad \perogni v \in \L2H, \quad \perogni T \in (0,+\infty).
  \eqno\duesette
$$
Moreover, we assume that
$$
  \eqalignno{
  & g \in L^1(0,\infty;H)
  &\dueotto
  \cr
  & \thetaz \in H, \qquad \chiz \in V.
  &\duenove
  \cr
  }
$$
\Def\HPdati{\accorpa\duesei\duenove}

A~weak formulation of problem \Ipbl\ can be stated as follows.

\let\etichetta\relax
\def\pbl{Problem~(P)}
\sect \pbl.
\sl
Find a pair $(\theta,\chi)$ such that, for each $T\in(0,+\infty)$,
$$
  \eqalignno{
  & \theta \in \W{1,1}{W'} \cap \L\infty H
  &\duedieci
  \cr
  & \chi \in \H1{V'} \cap \L\infty V \cap \L2W
  &\dueundici
  \cr
  & \eta := \theta + \ell \chi \in \C0H
  &\duedodici
  \cr
  & \<< \eta_t(t) , v >> - \intO (k*\theta)(t) \, \Delta v
    = \intO g(t) \, v
    \qquad \perogni v \in W, \quad
    \hbox{for a.e. } t \in (0,T)
    \qquad
  &\duetredici
  \cr
  & w := - \Delta \chi + \chi^3 - \chi - \ell \theta \in \L2V
  &\duequattordici
  \cr
  & \< \chi_t(t) , v > + \intO \nabla w(t) \cdot \nabla v = 0
    \qquad \perogni v \in V, \quad
    \hbox{for a.e. } t \in (0,T)
  &\duequindici
  \cr
  & \theta(0) = \thetaz, \qquad \chi(0) = \chiz.
  &\duesedici
  \cr
  }
$$

\bigskip
\rm
Let us comment on some of the above items.  Note that the homogeneous
Neumann boundary conditions~\Itre\ for $k*\theta$, $\chi$, and~$w$ are
contained in \duetredici, \dueundici, and~\duequindici, respectively.
From~\dueundici, by interpolation we have that $\chi\in\C0H$, whence
\duedodici\ yields $\theta\in\C0H$ as well, and the initial
conditions~\duesedici\ hold in~$H$.  Setting
$$
  (1*z)(t) = \int_0^t z(s) \, ds
  \qquad \hbox{for $t\in[0,T]$ and $z\in\L1{W'}$}
$$
an integration of \duetredici\ with respect to $t$ and the use of
\accorpa\dueundici\duedodici, \duesedici\ lead to the equality
$$
  \eqalignno{
  & \relax \<< \eta(t) , v >> - \intO (1*k*\eta)(t) \, \Delta v
    = \intO
      \Bigl(
        \thetaz + \ell \chiz
        + \bigl(
            1*g - \ell \Delta (1*k*\chi)
          \bigr) (t)
      \Bigr) v
    \qquad
  &
  \cr
  &\hskip.3\hsize \perogni v \in W, \quad \perogni t \in [0,T].
  &\duediciassette
  \cr
  }
$$
Therefore, in view of \accorpa\dueotto\duenove\ and~\dueundici, it turns
out~that
$$
  \eqalignno{
  & 1*k*\eta \in \C0W
  &\duediciottoA
  \cr
  & \eta(t) - \Delta (1*k*\eta)(t)
    = \thetaz + \ell \chiz + (1*g)(t) - \ell \Delta (1*k*\chi)(t)
    \qquad
  &
  \cr
  & \hskip.3\hsize \hbox{in $H$}, \quad \perogni t \in [0,T].
  &\duediciottoB
  \cr
  }
$$
\Def\duediciotto{\accorpa\duediciottoA\duediciottoB}

Concerning \duequattordici, we note that \dueundici\ and the 
continuous embedding of $H^1(\Omega)$ 
into $L^6(\Omega)$ (since $N\leq3)$ 
imply that $\chi^3\in\L\infty H$. Therefore,  from 
\accorpa\duedieci\dueundici\ it follows that $w\in\L2H$. 
The further regularity $w\in\L2H$ is then deduced from 
\duequindici, upon recalling that
$\chi_t\in\L2{V'}$. Finally, we point out~that
$$
  \eqalignno{
  \intO \chi(t) = \intO \chiz
  & \qquad \perogni t \in (0,+\infty)
  &\duediciannove
  \cr
  \intO \theta(t) = \intO \thetaz + \intQt g
  & \qquad \perogni t \in (0,+\infty).
  &\dueventi
  \cr
  }
$$
To  verify \duediciannove, it suffices to take $v=1$ in~\duequindici\ and
integrate in time.  Using~\duediciannove\ and~\duedodici, the same choice
in \duetredici\ yields~\dueventi.

\thm\Dueuno
There exists a unique solution $(\theta,\chi)$ to \pbl.  Moreover, the
estimate
$$
  \eqalignno{
  & \normaH{\theta(t)}^2
  + \int_0^t \normaVp{\chi_t(s)}^2 \, ds
  + \normaH{\nabla \chi(t)}^2
  &
  \cr
  & \leq C_1
    \tonde{
      1 + \normaH\thetaz^2 + \normaV\chiz^4 + \norma g_{\Lt1H}^2
    }
  &\dueventuno
  \cr
  }
$$
holds for any $t\in(0,+\infty)$, where $C_1$ is a positive constant
which depends only on~$\Omega$.\fine

The uniqueness result is the principal new contribution in the above
theorem and it is proved in the next section.  For the existence of
solutions to~(P) we refer to~[\NC].  In that paper, by employing
a Galerkin method for the approximation and deducing some a priori 
estimates, existence is shown by passing to the limit 
in the approximating scheme. Actually, the solutions found in [\NC]
are slightly less regular than those of~(P). Let us briefly describe 
the procedure. A basic uniform estimate is  formally obtained by taking 
$v=\eta(t)$ in~\duetredici\ and $v=\chi(t)$ in~\duequindici, adding
the two equalities, and controlling the various terms.  This
estimate yields that $\chi\in\L2W$.  Afterwards, an estimate similar to
\dueventuno\ is deduced, and a bound for the \lhs\ of \dueventuno\ is given
for intervals of type $(\delta,+\infty)$, $\delta>0$, exploiting the
regularizing effect of the evolution equation~\Idue.  In our setting, with
the initial data as in~\duenove, the regularizing effect is no longer
necessary and the gap between the results of [\NC] and the existence part
of Theorem~\Dueuno\ is easily closed using a suitable Gronwall
inequality (cf.~[\Brezis, Lemme~A4, p.156]) to deduce~\dueventuno, then
achieving~\dueundici.  The formal proof of \dueventuno\ proceeds by summing
\duetredici\ tested by $\theta(t)$ with \duequindici\ tested by
$\calN\chi_t(t)$ (note~that $\chi_t(t)\in\calV'$ by~\duequindici).  Two
troublesome terms (those containing~$\ell\chi_t\theta)$ cancel out.
Integrating by parts and also with respect to~$t$, remarking~that
$$
  \intQt (k*\nabla \theta)(s) \cdot \nabla \theta(s) \, ds
  \geq 0
$$
by virtue of~\duesette, and recalling \duedue\ and~\duequattro, we
infer~that
$$
  \displaylines{
  \unmezzo \normaH{\theta(t)}^2
    + \int_0^t \normaVp{\chi_t(s)}^2 \, ds
    + \unmezzo \normaH{\nabla \chi(t)}^2
    + {1 \over 4} \intO (\chi^2(t) - 1)^2
  \cr
  \leq \unmezzo \normaH\thetaz^2
    + \int_0^t \normaH{g(s)} \, \normaH{\theta(s)} \, ds
    + \unmezzo \normaH{\nabla \chiz}^2
    + {1 \over 4} \intO (\chizq - 1)^2.
  \cr
  }
$$
Now, it suffices to apply the same variant of Gronwall lemma noted above to
attain~\dueventuno.

As already stated in the Introduction, we are interested in investigating
the \longtime\ \bhv\ of the solution $(\theta,\chi)$ to \pbl.  In
particular, we wish to identify the cluster points of the trajectory
$(\theta(t),\chi(t))$ in a suitable topology.  With regard to \phf\ models
in general, a natural topology for the study of the asymptotic \bhv\ as
$t\toinfty$ is the strong topology of~$V'\times H$.  Thus, defining the
\omegalimit\ set associated with the initial data $\thetaz$ and $\chiz$~by
$$
  \omegaz =
  \graffe{
    \vcenter{\hsize.5\hsize\raggedright\tolerance1000 \noindent
      $(\thetai,\chii) \in V' \times H$ such that there exists
      a positive increasing sequence $\{t_n\}$, with $t_n\toinfty$
      and such that $\{(\theta(t_n),\chi(t_n))\}$ converges to
      $(\thetai,\chii)$ in $V' \times H$
    }
  },
  \eqno\dueventidue
$$
our result may be stated as follows.

\thm\Duedue
Assume that \HPdati\ hold and let $(\theta,\chi)$ be the corresponding
solution of \pbl.  Then the \omegalimit\ set $\omegaz$ is a nonempty,
compact, and connected subset of~$V' \times H$.  Moreover, if
$(\thetai,\chii)\in\omegaz$, then $\thetai$ assumes the constant value
$$
  \thetai
  = {1 \over |\Omega|}
  \tonde{
    \intO \thetaz + \intQinfty g
  }
  \eqno\dueventitre
$$
and $\chii\in V$ solves the problem
$$
  \eqalignno{
  - \Delta \chii + \chiic - \chii - \ell \thetai = w_\infty
  & \qquad\hbox{a.e.~in $\Omega$}
  &\dueventiquattroA
  \cr
  \intO \chii = \intO \chiz, \quad \partialn \chii = 0
  & \qquad\hbox{a.e.~on $\Gamma$}
  &\dueventiquattroB
  \cr
  }
$$
\Def\dueventiquattro{\accorpa\dueventiquattroA\dueventiquattroB}%
where $w_\infty$ is given by the constant
$$
  w_\infty = {1 \over |\Omega|} \intO (\chiic-\chii) - \ell \thetai.
  \eqno\dueventicinque
$$
Additionally, the entire family $\{\theta(t)\}$ converges in the sense~that
$$
  \theta(t) \to \thetai
  \quad \hbox{weakly in $H$ and strongly in $V'$ as $t\toinfty$}.\fine
  \eqno\dueventisei
$$

\rm
The proof of this theorem is given in Section~4.  At this point, we
comment on the structure of the set of solutions $\chii$ to the
stationary problem~\dueventiquattro, as Kenmochi, Niezg\'odka, and Pawlow
did in [\KNP,~Remark~6.1].  If we know, for instance, that the number of
solutions to \dueventiquattro\ is finite (as~in the one\_dimensional case
examined in~[\GriNov], where all the solutions are effectively specified)
we~could conclude in addition to~\dueventisei\ that $\chi(t)$ also has a
limit as $t\toinfty$.  Indeed, in this case the connectedness of $\omegaz$
would imply that $\omegaz$ is a singleton and the whole trajectory
$\{(\theta(t),\chi(t))\}$ converges.  In higher dimensions, some sufficient
conditions under which the cardinality of the set of stationary solutions
is finite have been given.  In this matter, we refer the reader to the
survey [\NST] and to references therein.

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

\parag\Continuousdependence
Continuous dependence

This section is devoted to the proof of uniqueness for \pbl\
(cf.~Theorem~\Dueuno).  More precisely, we prove the following continuous
dependence result

\lem\Amy
Let $k$ be as in \accorpa\duesei\duesette\ and let
$\{g_i,\theta_{0i},\chi_{0i}\}$, $i=1,2$, be two sets of data for \pbl\
which satisfy \accorpa\dueotto\duenove.  Denote by $(\theta_i,\chi_i)$,
$i=1,2$, two corresponding solutions and~put
$$
  M := \max_{i=1,2}
  \graffe{
    \normaVp{\theta_{0i}} + \normaVp{\chi_{0i}} + \norma{g_i}_{\L1{V'}}
  }.
  \eqno\treuno
$$
Then, for each $T\in(0,+\infty)$, there is a constant $C_2(T,M)$, which
also depends on $\norma k_{L^1(0,T)}$, $\Omega$, and~$\el$, such~that
$$
  \eqalignno{
  & \norma{\theta_1 - \theta_2}_{\C0{V'}}^2
    + \norma{\chi_1 - \chi_2}_{\C0{V'}}^2
    + \norma{\chi_1 - \chi_2}_{\L2V}^2
  &
  \cr
  & \leq C_2(T,M)
    \tonde{
      \normaVp{\theta_{01} - \theta_{02}}^2
      + \normaVp{\chi_{01} - \chi_{02}}^2
      + \norma{g_1 - g_2}_{\L1{V'}}^2
    }.\fine
  &\tredue
  \cr
  }
$$
\rm

In order to derive the estimate~\tredue, we need some basic tools from the
theory of Volterra integral differential equations.  Thus, referring to
[\Pruss] for details, we recall the following properties.

\prop\Tredue
Let the kernel $k$ satisfy \accorpa\duesei\duesette.  In addition, let
$X$~be a Hilbert space and let $A:D(A)\incluso X\to X$ be a linear, densely
defined, self\_adjoint, and positive semidefinite operator.  Given
$T\in(0,+\infty)$ and $z\in\W{1,1}X$, the equation
$$
  u(t) + A (1*k*u)(t) = z(t), \quad t\in[0,T]
  \eqno\tretre
$$
has a unique mild solution $u$; i.e., a function $u\in\C0X$, with
$1*k*u\in\C0{D(A)}$, fulfilling equation~\tretre\ for any $t\in[0,T]$.
Moreover, the solution $u$ of \tretre\ can be expressed~as
$$
  u(t)
  = S(t) z_0 + \int_0^t S(t-s) \, z'(s) \, ds, \quad t\in [0,T]
  \eqno\trequattro
$$
where $z'$ is the time derivative of~$z$.  Here $\{S(t),\ t\in[0,T]\}
\subset\calL(X)$ is the resolvent associated with $A$ and~$k$, which
provides a strongly continuous family of bounded linear operators on $X$
satisfying $S(0)=I$ (the~identity mapping), $\norma{S(t)}_{\calL(X)}\leq1$
and $S(t)D(A)\subset D(A)$ for any $t\in[0,T]$, and $AS(t)y=S(t)Ay$ for any
$t\in[0,T]$ and any $y\in D(A)$.\fine

In our setting, the above proposition is applied to equation~\duediciottoB.
Note that in this context we can choose $X=H$~and
$$
  A = - \Delta \quad \hbox{with} \quad D(A) = W.
$$
Alternatively, we can take $X=V'$, $D(A)=V$, and
$$
  \< A v_1 , v_2 > = \intO \nabla v_1 \cdot \nabla v_2,
  \qquad v_1, v_2 \in V.
  \eqno\trecinque
$$
Then, by the formulation of \pbl\ and \HPdati, it turns out that the
function $\eta$ is a mild solution of equation \duediciottoB\ both in the
spaces $H$ and~$V'$.  In particular, using \trequattro\ we have~that
$$
  \eta(t)
  = S(t) (\thetaz + \ell \chiz)
  + \int_0^t S(t-s) \, g(s) \, ds
  + \ell \int_0^t S(t-s) \, (k*A\chi)(s) \, ds
  \eqno\tresei
$$
for any $t\in[0,T]$, where $A$ is specified by \trecinque\ and where
$\{S(t),\ t\in[0,T]\} \subset\calL(V')$ is the related resolvent.

\proofof Lemma \Amy.
For the sake of convenience, here we use the same symbol for constants
which may vary from line to line.  We do not make explicit their
dependence on all the quantities specified in the statement and stress only
their dependence on $T$ and $M$ in the notation.  Moreover, we~set
$$
  \eta_i = \theta_i + \ell \chi_i, \qquad
  w_i = - \Delta \chi_i + \chi_i^3 - \chi_i- \ell \theta_i
$$
for $i=1,2$, and (within this proof)
$$
  \displaylines{
  \theta = \theta_1 - \theta_2, \qquad
  \chi = \chi_1 - \chi_2, \qquad
  \eta = \eta_1 - \eta_2, \qquad
  w = w_1 - w_2
  \cr
  g = g_1 - g_2, \qquad
  \thetaz = \theta_{01} - \theta_{02}, \qquad
  \chiz = \chi_{01} - \chi_{02}.
  \cr
  }
$$
In terms of this notation, we have that \duetredici,
\accorpa\duequindici\duesedici, \duediciannove, and \tresei\ still hold,
while \duequattordici\ for $\theta_i$, $\chi_i$, $w_i$, $i=1,2$, and
\duedodici\ imply~that
$$
  w
  = - \Delta \chi + \diffchic
  + (\el^2 - 1) \chi - \ell \eta
  \qquad \hbox{a.e.~in } \Omega \times (0,T).
  \eqno\tresette
$$
Using \tresei\ and the contractivity of~$S(t)$, we deduce that
$$
  \normaVp{\eta(t)}
  \leq \normaVp{\thetaz + \ell \chiz}
  + \int_0^t \normaVp{g(s)} \, ds
  + \ell \int_0^t \normaVp{(k*A\chi)(s)} \, ds
$$
for any $t\in[0,T]$.  From \trecinque\ it follows that
$\normaVp{(k*A\chi)(s)}\leq\normaH{(k*\nabla\chi)(s)}$ and consequently,
using the Young inequality for the convolution product, we obtain
$$
  \normaVp{\eta(t)}
  \leq \normaVp\thetaz
  + \ell \normaVp\chiz
  + \norma g_{\L1{V'}}
  + \ell \norma k_{L^1(0,T)} \int_0^t \normaH{\nabla \chi(s)} \, ds.
  \eqno\treotto
$$
Next, we observe that \duediciannove\ implies
$$
  {1 \over |\Omega|} \< \chi(t) , 1 >
  = \muz
  := {1 \over |\Omega|} \< \chiz , 1 >
  \qquad \perogni t \in [0,T]
$$
whence $\chi(t)-\muz\in\calV$ for a.e.~$t\in(0,T)$.  Therefore, we can
choose $v=\calN(\chi(t)-\muz)$ as test function in~\duequindici.  By
\duequattro\ and \duedue\ we have~that
$$
  \unmezzo {d \over dt} \intO |\nabla \calN (\chi(t) - \muz)|^2
  + \< \chi(t) - \muz , w(t) > = 0.
$$
Using \tresette, an integration by parts, and the obvious equality
$$
  \intO (\chi(t) - \muz) \, \chi(t)
  = \intO (\chi(t) - \muz)^2
$$
we easily obtain that
$$
  \displaylines{
  \unmezzo {d \over dt} \intO |\nabla \calN (\chi(t) - \muz)|^2
    + \intO |\nabla (\chi(t) - \muz)|^2
    + \intO
      \bigl(
        (\diffchic) \chi
      \bigr) (t)
  \cr
  + (\el^2 - 1) \intO (\chi(t)-\muz)^2
  = \muz \intO (\diffchic)(t)
    + \ell \< \eta (t) , \chi(t) - \muz >.
  \cr
  }
$$
Now, the obtained identity becomes
$$
  \eqalignno{
  & \unmezzo {d \over dt} \intO |\nabla \calN (\chi(t) - \muz)|^2
    + \intO |\nabla (\chi(t) - \muz)|^2
  &
  \cr
  & \qquad + \intO
      \bigl(
        (\diffchic) \chi
      \bigr) (t)
    + \el^2 \normaH{\chi(t)-\muz}^2
  &
  \cr
  & = \muz \intO (\diffchic)(t)
    + \<  \chi(t) - \muz + \ell \eta (t) , \chi(t) - \muz >
  &\trenove
  \cr
  }
$$
and we note that
$$
  \< (\chi - \muz + \ell \eta) (t) , \chi(t) - \muz >
  \leq C \normaVp{(\chi - \muz + \ell \eta)(t)}^2
  + \unmezzo \min \{ 1 , \el^2 \} \normaV{\chi(t) - \muz}^2.
$$
Thus, integrating \trenove\ with respect to $t$ and using \duequattro,
\duesedici, and the monotonicity of the function $y\mapsto y^3$, we
obtain~that
$$
  \eqalignno{
  & \normaVp{\chi(t) - \muz}^2
    + \int_0^t \normaV{\chi(s) - \muz}^2 \, ds
  &
  \cr
  & \leq C \normaVp{\chiz - \muz}^2
    + C_3 \muz \intQt ( \diffchic )
  &
  \cr
  & \qquad
    + C \int_0^t
      \tonde{
        \normaVp{\chi(s) - \muz}^2
        + \normaVp{\eta(s)}^2
      } \, ds
    \qquad \perogni t \in [0,T]
  &\tredieci
  \cr
  }
$$
where, in the notation, we have distinguished the constant $C_3$ from the
other~$C$\hskip-\mathsurround's.  We can now estimate the last term of
\tredieci\ with the help of~\treotto\ and the H\"older inequality.  We
deduce~that
$$
  \eqalignno{
  & \normaVp{\chi(t) - \muz}^2
    + \norma{\chi - \muz}_{\Lt2V}^2
  &
  \cr
  & \leq C \normaVp{\chiz - \muz}^2
    + C_3 \muz \intQt (\diffchic)
    + C T
      \tonde{
        \normaVp\thetaz^2
        + \normaVp\chiz^2
        + \norma g_{\L1{V'}}^2
      }
  &
  \cr
  & \qquad + C \int_0^t
    \tonde{
      \normaVp{\chi(s) - \muz}^2
      + T \norma k_{L^1(0,T)}^2 \norma{\chi - \muz}_{\Ls2V}^2
    } \, ds.
  &\treundiciL
  \cr
  }
$$
On the other hand, we have
$$
  \displaylines{
  C_3 \muz \intQt (\diffchic)
  \cr
  = C_3 \muz \intQt (\chi - \muz) (\chi_1^2 + \chi_2^2 + \chi_1 \chi_2)
  + C_3 \muz^2 \intQt (\chi_1^2 + \chi_2^2 + \chi_1 \chi_2)
  \cr
  \leq 2 C_3 |\muz| \intQt |\chi - \muz| (\chi_1^2 + \chi_2^2)
  + 2 C_3 \muz^2 \intQt (\chi_1^2 + \chi_2^2)
  \cr
  }
$$
which yields
$$
  \eqalignno{
  & C_3 \muz \intQt (\diffchic)
  &
  \cr
  & \leq \unmezzo \norma{\chi - \muz}_{\Lt2H}^2
  + C(T) \muz^2
    \tonde{
      1 + \normaLqQT{\chi_1}^4 + \normaLqQT{\chi_2}^4
    }. \qquad
  &\tredodiciL
  \cr
  }
$$
Therefore, combining \treundiciL\ with~\tredodiciL, we~get
$$
  \eqalignno{
  & \normaVp{\chi(t) - \muz}^2
    + \unmezzo \norma{\chi - \muz}_{\Lt2V}^2
  &
  \cr
  & \leq C \normaVp{\chiz - \muz}^2
    + C(T) \muz^2
    \tonde{
      1 + \normaLqQT{\chi_1}^4
      + \normaLqQT{\chi_2}^4
    }
  &
  \cr
  & \qquad + C(T)
    \tonde{
      \sqerror
    }
  &
  \cr
  & \qquad + C(T) \int_0^t
    \tonde{
      \normaVp{\chi(s) - \muz}^2
      + \norma{\chi - \muz}_{\Ls2V}^2
    } \, ds
  &\treimportante
  \cr
  }
$$
and we can apply the Gronwall lemma.  Writing $\chi=(\chi-\muz)+\muz$ and
noting~that
$$
  |\muz| \leq \normaVp\chiz
  \aand
  \normaVp{\chiz - \muz} \leq 2 \normaVp\chiz
$$
we conclude~that
$$
  \displaylines{
  \normaVp{\chi(t)}^2 + \norma\chi_{\Lt2V}^2
  \cr
  \leq C(T) \tonde{\sqerror}
  \cr
  + C(T) \normaVp\chiz^2
    \tonde{
      1 + \normaLqQT{\chi_1}^4
      + \normaLqQT{\chi_2}^4
    }.
  \cr
  }
$$
Hence, the estimate \tredue\ becomes a trivial consequence of the
forthcoming lemma.

\rem\Newremark
With reference to the above proof, note that the uniqueness part of
Theorem~\Dueuno\ follows by \treimportante, the Gronwall lemma, 
and \treotto, since $\mu_0=0$ in this case. Morevorer, whenever
$\muz=0$,~i.e.,
$$
  \< \chi_{01}, 1 > = \< \chi_{02}, 1 >
$$
the estimate~\tredue\ turns out to hold with a constant $C_2(T)$ which does
not depend on~$M$, as one can immediately see taking $\muz=0$
in~\treimportante.

\rem\Tretre
We notice that the techniques developed in this section are similar to
those employed in [\NC] and [\BHN] for the conserved \phf\ equation with
and without memory, respectively, and to those employed in [\CLdue] for
the nonconserved \phf\ problem with memory.

\lem\Philippe
Assume that \accorpa\duesei\duenove\ hold and let $(\theta,\chi)$ be the
corresponding solution to \pbl.  Then, for any $T\in(0,+\infty)$, there
exists a constant $C_4(T)$ such~that
$$
  \normaLqQT\chi^4
  \leq C_4(T)
    \tonde{
      1 + \normaVp\chiz^4
      + \normaVp\thetaz^2
      + \norma g_{\L1{V'}}^2
    }.
  \eqno\philippe
$$
Moreover, the constant $C_4(T)$ depends only on $T$, $\norma k_{L^1(0,T)}$,
$\Omega$, and~$\el$.

\proof
We set $\muz:=|\Omega|^{-1}\<\chiz,1>$ and note that the function
$$
  t \mapsto {1 \over |\Omega|} \< \chi(t) , 1 >
$$
takes the constant value $\muz$ by~\duediciannove.  Then, choosing
$v=\calN(\chi(t)-\muz)$ in~\duequindici, we obtain
$$
  \displaylines{
  \unmezzo {d \over dt} \intO |\nabla \calN (\chi(t) - \muz)|^2
  + \intO |\nabla (\chi(t) - \muz)|^2
  + \intO
    \bigl(
      (\chi^3 - \chi) (\chi - \muz)
    \bigr) (t)
  \cr
  = \ell \intO \bigl( \eta (\chi - \muz) \bigr) (t)
  - \el^2 \intO (\chi - \muz)^2 (t)
  \leq \ell \normaVp{\eta(t)} \normaV{\chi(t) - \muz}
  - \el^2 \normaH{\chi(t) - \muz}^2
  \cr
  \leq \delta \normaV{\chi(t) - \muz}^2
  + {\el^2 \over 4 \delta} \normaVp{\eta(t)}^2
  - \el^2 \normaH{\chi(t) - \muz}^2
  \cr
  }
$$
where $\delta=(1/2)\min\{1,\el^2\}$.  Thus, we deduce that
$$
  \displaylines{
  \unmezzo {d \over dt} \intO |\nabla \calN (\chi(t) - \muz)|^2
  + \delta \normaV{\chi(t) - \muz}^2
  \cr
  + \intO \chi^4 (t) - \muz \intO \chi^3(t)
  - \intO \chi^2 (t) + \muz^2 |\Omega|
  \leq {\el^2 \over 4 \delta} \normaVp{\eta(t)}^2.
  \cr
  }
$$
Using the elementary Young inequality and integrating in time, it is \sfw\
to find a constant $C_5$ such~that
$$
  \norma{\chi - \muz}_{\Lt2V}^2
  + \intQt \chi^4
  \leq C_5
    \tonde{
       \normaVp{\chiz-\muz}^2 + 1 + \muz^4
       + \int_0^t \normaVp{\eta(s)}^2 \, ds
    }.
$$
On the other hand, following the proof of~\treotto, it is easy to
demonstrate~that
$$
  \normaVp{\eta(t)}
  \leq \normaVp\thetaz + \ell \normaVp\chiz + \norma g_{\L1{V'}}
  + \ell \norma k_{L^1(0,T)} \norma{\chi - \muz}_{\Lt1V}.
$$
Therefore, we immediately obtain \philippe\ by combining the two last
inequalities, estimating $|\muz|$ by~$\normaVp\chiz$, and applying the
Gronwall lemma.

\rem\Trequattro
Let us point out that Lemma~\Amy\ gives us furthermore the 
possibility of defining yet a weaker solution of the problem 
\Ipbl\ when the data $g, \, \thetaz , \, \chiz$ are only 
assumed to be in $\L1{V'}, \  V', \  V'\,,$ respectively.  Indeed,
a consequence of the estimate \tredue\ is that it would suffice to
approximate $g, \, \thetaz, \, \chiz$ in the above norms by $g_\eps,
\, \theta_{0\eps}, \, \chi_{0\eps}$ such that
$$
  g_\eps \in \L1H, \qquad
  \theta_{0\eps} \in H, \aand
  \chi_{0\eps} \in V
  \eqno\tredodici
$$
for any $\eps>0$ (cf.~\accorpa\dueotto\duenove), then argue by Cauchy
sequences, and select the limit as $\eps\tozero$.  In this case we would
obtain a solution $(\theta,\chi)$~with
$$
  \theta \in \C0{V'}
  \aand
  \chi \in \C0{V'} \cap \L2V.
$$
Moreover, this solution would then be unique in the class of solutions
obtained as limits of sequences of weak solutions (in~the sense of \pbl),
with the approximating data which satisfy~\tredodici, and would depend
continuously on the~data.

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

\parag\Longtimbhv
\Longtime\ \bhv{}

In this section we present the proof of Theorem~\Duedue.  First of all,
note that by Theorem~\Dueuno\ and, in particular, by estimate~\dueventuno\
(coupled with \duediciannove\ and the Poincar\'e inequality) the entire set
$\{(\theta(t),\chi(t)),\ t\geq0\}$ is bounded in $H\times V$ and relatively
compact in $V'\times H$.  Therefore, the set $\omegaz$ is a nonempty
compact subset of $V'\times H$.  Actually, $\omegaz$ is also connected, due
to the continuity of $(\theta,\chi)$ from $[0,+\infty)$ to $V'\times H$ and
to a standard argument from the theory of dynamical systems (one~can see,
for instance, [\Haraux,~p.~12]).

Now, let $(\thetai,\chii)\in\omegaz$ and fix a strictly increasing sequence
$\graffe{t_n}_{n\geq1}$ of positive real numbers such that $t_n\toinfty$ as
$\ntoinfty$~and
$$
  \theta(t_n) \to \thetai
  \quad \hbox{in } V'
  \aand
  \chi(t_n) \to \chii
  \quad \hbox{in } H.
  \eqno\quattrouno
$$
Furthermore, we introduce the functions
$$
  \psi := \theta - \thetaz - 1*g, \qquad
  \xi := k*\theta, \aand
  \ffi := k*\psi.
  \eqno\defnuove
$$
Recall also that $w=-\Delta\chi+\chi^3-\chi-\ell\theta$.  The following
preliminary results summarize some properties which are useful for the
sequel.

\lem\Quattrouno
If $(\thetai,\chii)\in\omegaz$, then
$$
  \intO \chii = \intO \chiz
  \aand
  \intO \thetai = \intO \thetaz + \intQinfty g.\fine
  \eqno\quattrodue
$$

\proof
This is a \sfw\ consequence of \quattrouno\ and
\accorpa\duediciannove\dueventi.

\lem\Quattrodue
We have that
$$
  \eqalignno{
  & \theta \in \LL\infty H, \quad
    \xi \in \LL\infty H
  &\quattrotre
  \cr
  & \chi \in \LL\infty V, \quad
    \chi_t \in \LL2{V'}
  &\quattroquattro
  \cr
  & \psi \in \LL\infty H, \quad
    \psi_t \in \LL2{V'} + \LL\infty{W'}
  &\quattrocinque
  \cr
  & \ffi \in \LL\infty H, \quad
    \ffi_t \in \LL2{V'} + \LL\infty{W'}
  &\quattrosei
  \cr
  & w \in \LL\infty {V'}.\fine
  &\quattrosette
  \cr
  }
$$

\proof
To check \accorpa\quattrotre\quattroquattro\ it is sufficient to recall
\dueventuno, \duesei, \dueotto, \duediciannove\ and to note~that
$$
  \norma\xi_{\L\infty H}
  \leq \norma k_{L^1(0,T)} \norma\theta_{\L\infty H}
  \qquad \perogni T \in (0,+\infty).
$$
Moreover, \quattrocinque\ follows from \quattrotre,
\accorpa\dueotto\duenove, and from \accorpa\duedodici\duetredici, which
imply~that
$$
  \<< \psi_t (t) , v >>
  = - \ell \<< \chi_t (t) , v >> + \intO \xi(t) \, \Delta v
  \qquad \perogni v \in W, \quad \hbox{for a.e.~$t\in(0,+\infty)$}.
$$
This means
$$
  \psi_t (t) = - \ell \chi_t (t) + B \xi(t)
  \qquad \hbox{in $W'$}, \quad \hbox{for a.e.~$t\in(0,+\infty)$}
  \eqno\quattrootto
$$
where $B\in\calL(H;W')$ is defined by
$$
  \<< B u , v >> = \intO u \, \Delta v
  \qquad \perogni u \in H, \quad \perogni v \in W.
  \eqno\defB
$$

Incidentally, let us note that $Bu=\Delta u$ only for particular
$u$\kern-\mathsurround's.  Indeed, we have~that
$$
  \eqalignno{
  \<< B u , v >>
    = - \intO \nabla u \cdot \nabla v
  & \qquad \perogni u \in V, \quad \perogni v \in W
  &
  \cr
  \<< B u , v >>
    = \intO (\Delta u) \, v - \int_\Gamma (\partialn u) \, v
  & \qquad \perogni u \in H^2(\Omega), \quad \perogni v \in W
  &
  \cr
  \<< B u , v >>
    = \intO (\Delta u) \, v
  & \qquad \perogni u \in W, \quad \perogni v \in W
  &
  \cr
  }
$$
and the last formula can be rewritten as
$$
  B u \in H \and B u = \Delta u
  \qquad \perogni u \in W.
  \eqno\propB
$$
Moreover, we note that
$$
  u \in H \and B u = 0
  \qquad \hbox{imply} \qquad
  u = \hbox{constant}.
  \eqno\perdopo
$$
Indeed, for any $\zzz\in\calD(\Omega)^N$, the mean value of $\div\zzz$
vanishes; hence, we can find $v\in W$ such that $-\Delta v=\div\zzz$, use
it in~\defB, and deduce $\nabla u=0$ in the sense of distributions.

On the other hand, one can easily verify that
$$
  \ffi = \xi - k*(\thetaz + 1*g) = k*\psi
$$
and by~\duesedici,
$$
  \xi_t = k \thetaz + k*\theta_t, \qquad
  \ffi_t = k*\psi_t.
  \eqno\quattronove
$$
Recalling again~\duesei, the regularity properties in \quattrosei\ follow
from \quattrocinque\ and from the Young inequality for convolution
products.  Finally, \quattrosette~is derived simply using
\accorpa\quattrotre\quattroquattro, $V\subset L^6(\Omega)$, and
$-\Delta\chi\in\LL\infty {V'}$.  This concludes the proof.
\bigskip

Next, for $t\geq0$ and integers $n\geq1$ we define the auxiliary functions
$$
  \eqalignno{
  & \thetan(t) := \theta(t_n + t), \qquad
    \xin(t) := \xi(t_n + t), \qquad
    \chin(t) := \chi(t_n + t)
  &
  \cr
  & \psin(t) := \psi(t_n + t), \qquad
    \ffin(t) := \ffi(t_n + t), \qquad
    \wn(t) := w(t_n + t).
  &
  \cr
  }
$$
We are interested in studying the limiting \bhv\ of the above sequences as
$\ntoinfty$, in all finite intervals~$[0,T]$.  We remark that some
inconvenience is caused by the fact that $\xin\not=k*\thetan$.  It will
become clear in the analysis which follows how this difficulty is overcome.

\lem\Quattrotre
For any $T\in(0,+\infty)$ there is a constant $C_6(T)$ such~that
$$
  \eqalignno{
  & \norma\chin_{\L\infty V}
    + \norma{\partialt\chin}_{\L2{V'}}
    \leq C_6(T)
  &\quattrodieci
  \cr
  & \norma\wn_{\L2V}
    + \norma\chin_{\L2W}
    \leq C_6(T)
  &\quattroundici
  \cr
  & \norma\thetan_{\L\infty H}
    + \norma\xin_{\L\infty H}
    \leq C_6(T)
  &\quattrododici
  \cr
  & \norma\psin_{\L\infty H}
    + \norma{\partialt\psin}_{\L2{W'}}
    \leq C_6(T)
  &\quattrotredici
  \cr
  & \norma\ffin_{\L\infty H}
    + \norma{\partialt\ffin}_{\L2{W'}}
    \leq C_6(T).\fine
  &\quattroquattordici
  \cr
  }
$$

\proof
We start showing the indicated regularity for $\chin$ and~$\wn$. From
\accorpa\duequattordici\duequindici\ we have~that
$$
  \< \partialt \chin(t) , v >
  + \intO \nabla \wn(t) \cdot \nabla v
  = 0
  \qquad \perogni v \in V, \quad \hbox{for a.e.~$t\in(0,T)$}
  \eqno\quattroquindici
$$
with
$$
  \wn = - \Delta \chin + \chinc - \chin - \ell \thetan.
  \eqno\quattrosedici
$$
Using \quattroquattro, we readily infer \quattrodieci\ and a comparison in
\quattroquindici\ yields that $\nabla\wn$ is bounded
in~$\L2{L^2(\Omega;\erre^N)}$ independently of~$n$.  Now, from
\quattrosette\ we have that $|\<\wn,1>|$ is uniformly bounded.  Employing
the Poincar\'e inequality, we obtain the first part of~\quattroundici,
while the second part results from~\quattrosedici, using
\accorpa\quattrotre\quattroquattro, $V\subset L^6(\Omega)$, and the
boundary condition $\partialn\chin=0$ on $\Gamma\times(0,T)$.
Concerning \accorpa\quattrododici\quattroquattordici, it suffices to
recall \quattrotre\ and \accorpa\quattrocinque\quattrosei.  This concludes
the proof of the lemma.

\lem\Quattroquattro
For any $T\in(0,+\infty)$, as $\ntoinfty$ the following
convergences~hold
$$
  \eqalignno{
  \chin \to \chii
  & \quad \hbox{strongly in $\C0H$ and weakly star in $\L\infty V$}
  &\quattrodiciotto
  \cr
  \partialt\chin \to 0
  & \quad \hbox{strongly in $\L2{V'}$}.\fine
  &\quattrodiciannove
  \cr
  }
$$

\proof
In view of \quattroquattro, we have that
$$
  \norma{\partialt\chin}_{\L2{V'}}
  \leq \norma{\partialt\chi}_{L^2(t_n,\infty;V')}
  \to 0
$$
as $\ntoinfty$.  Therefore, \quattrodiciannove\ is verified and
$$
  \displaylines{
  \normaVp{\chin(t) - \chii}
    \leq \normaVp{\chin(t) - \chin(0)}
    + \normaVp{\chi(t_n) - \chii}
  \cr
  \leq \int_0^t \normaVp{\partialt \chin(s)} \, ds
    + \normaVp{\chi(t_n) - \chii}
  \cr
  }
$$
for all $n\geq1$ and $t\in[0,T]$.  Then applying the H\"older inequality
and~\quattrodiciannove, one infers from \quattrouno\ that $\chin\to\chii$
in $\C0{V'}$ (recall that $\chii$ is independent of time).
Using~\quattrodieci, a standard weak star compactness argument implies the
second convergence in~\quattrodiciotto.  Thus $\chii\in V$.  Therefore the
interpolation inequality
$$
  \normaH{\chin(t) - \chii}^2
  \leq \normaVp{\chin(t) - \chii} \normaV{\chin(t) - \chii}
$$
valid for $n\geq1$ and $t\in[0,T]$, enables us to complete the proof
of~\quattrodiciotto.

\lem\Quattrocinque
The limits $\chii$ and $\thetai$ are in $W$ and~$H$, respectively.
Moreover, setting
$$
  \wi :=
  {1 \over |\Omega|} \intO (\chiic - \chii - \ell \thetaz)
  - {\el \over |\Omega|} \intQinfty g
  \eqno\quattroventi
$$
$\chii$, $\thetai$, $\wi$ satisfy \dueventiquattro\ and, for each
$T\in(0,+\infty)$,
$$
  \eqalignno{
  \chin \to \chii
  & \qquad \hbox{weakly in $\L2W$ and strongly in $\L2V$}
  &\quattroventuno
  \cr
  \wn \to \wi
  & \qquad \hbox{weakly in $\L2V$}
  &\quattroventidue
  \cr
  \thetan \to \thetai
  & \qquad
    \hbox{weakly star in $\L\infty H$ and strongly in $\C0{W'}$}.\fine
    \qquad
  &\quattroventitre
  \cr
  }
$$

\proof
Let us first note that \quattroventuno\ is a consequence of \quattroundici,
\quattrodiciotto, and an interpolation inequality.  From
\accorpa\quattrodieci\quattrotredici, it follows that there exist
$\zetab,\wb,\thetab,\psib$ and a subsequence of 
$(\chi_n,w_n,\theta_n,\psi_n)$ (not relabeled) such that
$$
  \eqalignno{
  \chinc \to \zetab
  & \qquad \hbox{weakly star in $\L\infty H$}
  &\quattroventiquattro
  \cr
  \wn \to \wb
  & \qquad \hbox{weakly in $\L2V$}
  &\quattroventicinque
  \cr
  \thetan \to \thetab
  & \qquad \hbox{weakly star in $\L\infty H$}
  &\quattroventisei
  \cr
  \psin \to \psib
  & \qquad \hbox{weakly star in $\L\infty H$}
  &\quattroventisette
  \cr
  \partialt\psin \to \partialt\psib
  & \qquad \hbox{weakly in $\L2{W'}$}.
  &\quattroventotto
  \cr
  }
$$
The weak convergence \quattroventiquattro, the strong convergence in
\quattrodiciotto, and a standard argument for maximal monotone operators
(see, e.g., [\Barbu,~Lemma~1.3, p.~42]) imply that $\zetab=\chiic$.  We
note that the same conclusion can be achieved via almost everywhere
convergences deduced from~\quattrodiciotto, taking into account also 
\quattroventiquattro\ and applying [\Lions,~Lemme~1.3, p.~12]). Then, 
passing to the (weak) limit in~\quattrosedici, we find~that
$$
  \wb = - \Delta \chii + \chiic - \chii - \ell \thetab
  \eqno\quattroventinove
$$
while \quattroquindici, \quattrodiciannove, and \quattroventicinque\ yield
$$
  \intQT \nabla \wb \cdot \nabla v = 0
  \qquad \perogni v \in \L2V.
$$
Taking $v=\wb$ entails $\nabla\wb=0$, whence it turns out that, for 
a.e.~$t\in(0,T)$,  $\wb(t)$~is a constant function
in~$\Omega$, and $\wb\in L^2(0,T)$ because of~\quattroventicinque.
By~\quattroventicinque\ we also have~that
$$
  \wb \quad \hbox{is the weak limit of} \quad
  {1 \over |\Omega|} \< \wn , 1 >
  \quad \hbox{in $L^2(0,T)$}.
$$
By virtue of \quattrosedici, \quattroventuno, \quattroventiquattro,
\dueventi, and
$$
  \< \thetan(t) , 1 >
  = \intO \thetaz + \intQtn g
  \qquad \perogni t \in (0,\infty)
$$
it is not difficult to deduce that (cf.~\quattroventi)
$$
  {1 \over |\Omega|} \< \wn , 1 > \to \wi
  \qquad \hbox{weakly in $L^2(0,T)$}.
$$
Therefore, we conclude that $\wb=\wi$ and $\wi$ is independent of time
by~\quattroventi.  Moreover, \quattroventidue\ is plainly recovered
from~\quattroventicinque, the convergence holding for the whole sequence
because the limit is uniquely identified.  Due to~\quattroventinove, the
function $\thetab$ is constant with respect to time.  It still remains to
prove \quattroventitre.  In view of
\accorpa\quattroventisette\quattroventotto, the Ascoli compactness lemma
allows us to infer
$$
  \psin \to \psib
  \qquad \hbox{strongly in $\C0{W'}$}.
$$
As $\psin=\thetan-\thetaz-(1*g)(t_n+\cdot)$, using \dueotto\ and
\accorpa\quattroventisei\quattroventisette\ one easily checks~that
$$
  \psib = \thetab - \thetaz - \int_0^\infty g(s) \, ds
  \aand
  \thetan \to \thetab
  \quad \hbox{strongly in $\C0{W'}$}.
$$
Recalling that $\thetab$ does not depend on~$t$, this last convergence and
\quattrouno\ imply
$$
  \thetab = \thetab(0) = \lim_{\ntoinfty} \theta(t_n) = \thetai
$$
and (cf.~\quattroventisei) \quattroventitre\ is demonstrated.
Finally, let us remark that \quattroventotto\ reduces to
$$
  \partialt \psin \to 0
  \qquad \hbox{weakly in $\L2{W'}$}
  \eqno\quattrotrenta
$$
and holds for the entire sequence.

\lem\Quattrosei
There exist a function $\xib\in\C0{W'}$ and a subsequence of~$\{n\}$ (still
denoted by~$\{n\})$ such that, for each $T\in(0,+\infty)$,
$$
  \eqalignno{
  & \xin \to \xib \qquad
    \hbox{weakly star in $\L\infty H$ and strongly in $\C0{W'}$}
    \qquad
  &\quattrotrentuno
  \cr
  & \lim_{t\toinfty} \normaWp{\xib(t) - (k*\thetai)(t)} = 0.\fine
  &\quattrotrentadue
  \cr
  }
$$

\proof
Owing to \quattrododici\ and \quattroquattordici, by a diagonal procedure
we can select a subsequence $\ntoinfty$ such that there exist functions
$\xib,\ffib:(0,+\infty)\to W'$ satisfying
$$
  \eqalignno{
  \xin \to \xib \and \ffin \to \ffib
  & \qquad \hbox{weakly star in $\L\infty H$}
  &
  \cr
  \partialt\ffin \to \partialt\ffib
  & \qquad \hbox{weakly in $\L2{W'}$}
  &
  \cr
  }
$$
for all $T\in(0,+\infty)$.  Arguing as in the above proof, one
verifies~that
$$
  \ffib
  = \xib - \thetaz \int_0^\infty k(s) \, ds
  - \int_0^\infty (k*g)(s) \, ds
$$
and deduces the strong convergence in~\quattrotrentuno.  In order to
check~\quattrotrentadue, we estimate the norm
$\normaWp{\xib(t)-(k*\thetai)(t)}$ in the following manner
(cf.~[\CLuno,~Lemma~4.5])
$$
  \eqalignno{
  & \normaWp{(\xib - k*\thetai) (t)}
  &
  \cr
  & \leq \normaWp{\xib(t) - \xin(t)}
    + \normaWp{\xin(t) - (k*\thetan)(t)}
    + \normaWp{(k*(\thetan - \thetai))(t)}
    \qquad
  &\quattrotrentatre
  \cr
  }
$$
with (this being the crucial step)
$$
  \displaylines{
  \normaWp{\xin(t) - (k*\thetan)(t)}
  \cr
  =
  \normaWp{
    \int_0^{t_n + t} k(s) \, \theta(t_n + t - s) \, ds
    - \int_0^t k(s) \, \theta(t_n + t - s) \, ds
  }
  \cr
  \leq \int_t^{t_n + t} \normaWp{k(s) \, \theta(t_n + t - s)} \, ds
  \leq \norma\theta_{\LL\infty{W'}} \int_t^\infty |k(s)| \, ds
  \cr
  }
$$
and
$$
  \normaWp{(k*(\thetan - \thetai))(t)}
  \leq \norma k_{L^1(0,t)} \norma{\thetan - \thetai}_{\Ct0{W'}}.
$$
Then, on account of \quattrotrentuno, \quattrotre, and \quattroventitre,
allowing $\ntoinfty$ in \quattrotrentatre\ we~find
$$
  \normaWp{\xib(t) - (k*\thetai)(t)}
  \leq \norma\theta_{\LL\infty{W'}} \int_t^\infty |k(s)| \, ds
$$
for any $t\in(0,+\infty)$, and \quattrotrentadue\ now follows easily.

\lem\Quattrosette
The function $\xib$ defined by Lemma~\Quattrosei\ is a function of time
only.  Moreover, $\thetai$~is a constant and is specified
by~\dueventitre.\fine

\proof
Writing \quattrootto\ for $\psin,\chin,\xin$ and passing to the limit
as~$\ntoinfty$, by \quattrodiciannove\ and
\accorpa\quattrotrenta\quattrotrentuno\ we obtain
$$
  B \xib(t) = 0
  \qquad \hbox{in $W'$}, \quad \hbox{for a.e.~$t\in(0,+\infty)$}
$$
where $B$ is defined by~\defB.  Hence, the function $\xib(t)$ is constant
in~$\Omega$ by~\perdopo, being also a continuous function of~$t$.  Next, as
$\thetai\in H$ is independent of time, we~have
$$
  \<< \xib(t) - (k*\thetai)(t) , v >>
  = - (\thetai , v) \int_0^t k(s) \, ds
$$
for all $v\in\calW$.  Now, \quattrotrentadue\ enables us to conclude that
$$
  0 = - (\thetai , v) \int_0^\infty k(s) \, ds
  \qquad \perogni v \in \calW.
$$
Since the above integral differs from $0$ because of~\duesei, by density we
obtain that $(\thetai,v)=0$ for all $v\in\calH$, and this implies that
$\thetai$ is constant.  Finally, \dueventitre\ plainly follows
from~\quattrodue\ and the proof of the lemma is complete.\fine

In conclusion, we note that, collecting the results of
Lemma~\Quattrocinque\ and Lemma~\Quattrosette, Theorem~\Duedue\ is proved.

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$$\vbox{\everycr{\noalign{\smallskip}}%
  \halign{\smallfont#\hfil\qquad&\email#\hfil\cr
  \multispan2\hfil\bfs E--mail addresses of the authors\hfil\cr
  \noalign{\medskip}
  Pierluigi Colli       & pier@dragon.ian.pv.cnr.it    \cr
  Gianni Gilardi        & gilardi@dragon.ian.pv.cnr.it \cr
  Philippe Lauren\c cot & laurenco@iecn.u-nancy.fr     \cr
  Amy Novick--Cohen     & amync@tx.technion.ac.il      \cr
}}$$

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