Asymptotic Behaviour of Solutions to Phase Field Models with ...

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_{*}