In the context of solid-liquid phase transitions, $Â¥theta:=Â¥rho(u)$ ...... is a non-constant solution of $P(u^{Â¥infty})$, hence $vÂ¥inÂ¥tilde{S}_{0}$ . $Â¥phi$ ..... }^{l})x_{L}^{l},x_{R}^{l})iifflliissoevendd,,ã»ã»v=Â¥{iiffÂ¥iota_{l}iisseoddven,,Â¥end{array}Â¥right.
Funkcialaj Ekvacioj, 39 (1996) 123-142
Asymptotic Behaviour of Solutions to Phase Field Models with Constraints By
Akio ITO and Nobuyuki KENMOCHI (Chiba University, Japan)
Abstract. A phase field model with constraints is studied. The model is described as a coupled system of nonlinear parabolic PDEs governed by the absolute temperature and the order parameter. One of the PDEs includes a maximal monotone graph as a mathematical expression of a certain kind of constraints for the order parameter. In this paper we restrict the space dimension to one, and investigate . the structure of the -limit set of any solution of the model as time $t$
$¥mathrm{c}¥mathrm{o}$
1.
$¥rightarrow+¥infty$
Introduction
We consider a model for solid-liquid phase transitions of the following form:
(1.1)
in
$[¥rho(u)+¥lambda(w)]_{t}-u_{xx}=f(t, x)$
$Q:=(0, +¥infty)¥times(-L, L)$
(1.2)
$w_{t}-¥kappa w_{xx}+¥xi+g(w)=¥lambda^{¥prime}(w)u$
(1.3)
$¥xi¥in¥partial I_{[¥sigma_{*},¥sigma^{*}]}(w)$
(1.4)
$-u_{x}(t, -L)+n_{0}u(t, -L)=h_{¥_}(t)$
(1.5) (1.6)
,
in
,
increasing function of , and $u$
,
$Q$
for
$¥lambda(w)$
,
$¥kappa>0$
and
$t>0$
for
$w(0, ¥mathrm{x})=w_{0}(x)$
Here $L>0$ is a positive number;
,
$Q$
for
$u_{x}(t, L)+n_{0}u(t, L)=h_{+}(t)$
$w_{x}(t, -L)=w_{X}(t, L)=0$ $¥mathrm{u}(0, x)=¥mathrm{u}_{0}(x)$
in
$n_{0}>0$
$¥lambda^{¥prime}(w)=¥frac{d}{dw}¥lambda(w)$
,
,
$x$
$>0$
,
,
$¥in(-L, L)$
.
are constants;
$g(w)$
$t$
are
$C^{2}-$
$¥rho(u)$
is an
functions of
$w$
;
of the interval is the subdifferential of the indicator function this system given data. We call are and , , “Phase-Field Model with Constraints” and denote it by (PFC). represents the In the context of solid-liquid phase transitions, the order parameter which indicates the physical absolute temperature and is assumed to be range of the order parameter situation of the system; the , and $w(t, x)=¥sigma_{*}$ and $w(t, x)=¥sigma^{*}$ mean respectively a compact interval $I_{[¥sigma_{*},¥sigma^{*}]}$
$¥partial I_{[¥sigma_{*},¥sigma^{*}]}$
$[¥sigma_{*}, ¥sigma^{*}]¥subset R;f(t, ¥mathrm{x})$
$h_{¥pm}(t)$
$u_{0}$
$w_{0}$
$¥theta:=¥rho(u)$
$w$
$w$
$[¥sigma_{*}, ¥sigma^{*}]$
124
Akio ITO and Nobuyuki KENMOCHI
that the physical situation at $(t, x)$ is of pure solid and pure liquid, while $¥sigma_{*}