Asymptotic behaviour of solutions to non-isothermal phase separation model with constraint in one-dimensional space
Akio Ito, Nobuyuki Kenmochi · Journal of the Mathematical Society of Japan · 1998
Let us consider a one-dimensional model for non-isothermal phase separation, which is given by the following system, denoted by (PSC):$[\rho(u)+\lambda(w)]_{t}-u_{XX}=f(t, x)$ in $Q:=(0, +\infty)\cross J$ , (1.1) $w_{t}-\{-\kappa w_{XX}+\xi+g(w)-\lambda'(w)u\}_{xx}=0$$[-\kappa w_{XX}(t, \cdot)+\xi(t, \cdot)+g(w(t, \cdot))-\lambda'(w(t, \cdot))u(t, \cdot)]_{\chi}|_{x=\pm L}=0$ for $t>0$ , (1.5-2)(1.6)Here $J:=(-L, L)$ with a positive number $L;k>0$ and $n_{0}>0$ are constants; $\rho(u)$ is an increasing function of $u$ , and $\lambda(w),$ $\lambda'(w)=(d/dw)\lambda(w),$ $g(w)$ are smooth functions of $w$ ;is the subdifferential of the indicator function $I_{[\sigma_{*},\sigma^{*}]}$ of the interval $[\sigma_{*}, \sigma^{*}]\subset R$ ;$f(t, x),$ $h_{\pm}(t),$ $u_{0}(x)$ and $w_{0}(x)$ are given data.The above system arises in the phase separation of a binary mixture with components A and B. In this context, $\theta:=p(u)$ represents the absolute temperature and $w:=w_{A}$ the order parameter which is the local concentration of the component $A$ ; note that $\sigma_{*}=0\leq w_{A}(t, x)\leq 1=\sigma^{*}$ , and $w_{A}(t, x)=1$ (resp.$w_{A}(t,$ $x)=0$) means that the phase (the physical situation of the system) at $(t, x)$ is of pure A (resp.pure B), while $0<w_{A}(t,x)<1$ means that the phase at $(t, x)$ is of mixture.Along the same approach as $[1, 13]$ , the system $(1.1)-(1.3)$can be derived from a free energy functional of Landau- Ginzburg type $F_{\Omega}( \theta;w):=\int_{J}\{\frac{\kappa\theta}{2}|w_{X}|^{2}+\tau(\theta)+\theta(I_{[0,1]}(w)+\hat{g}(w))+\lambda(w)\}dx$ for $w\in H^{1}(J)$ , $where\hat{g}isaprimitiveofgand\tau(\theta)isasmoothfunctionof\theta satisfying\theta=\tau(\theta)-\theta\tau'(\theta)$ $(=p(u))$ .where $m_{0}:= \int_{J}w_{0}dx$ .The purpose of the present paper is to investigate the structure of the solution set of $P(\sigma_{*}, \sigma^{*}; u_{\infty},m_{0})$ and further some common properties of the $\omega$ -limit points of $w$ .In Shen & Zheng [14], the problem without constraint (1.3) was independently studied.They proved existence, uniqueness and asymptotic convergence of the solution.AS far as the asymptotic behaviour of the solution as $tarrow+\infty$ is concemed, our situation is much more complicated than theirs.Indeed, in our case, the order parameter $w(t, x)$ does not asymptotically converge and may oscillate as $tarrow+\infty$ , although it is very slow in time; this might come from constraint (1.3).In particular, when the temperature $\theta=\rho(u)$ is supposed to $h\vee e$ constant (hence $u$ to be constant), system $(1.2)-(1.3)$ is called "Cahn-Hilliard model with constraint", which was treated so far in $[2, 12]$ .This model was introduced as the quench limit of temperature $\theta\downarrow 0$ and studied in the case of $g(w)-\lambda'(w)u\equiv-cw$ with a positive constant $c$ by Blowey&Elliott [2] and a more general case by Kenmochi, Niezg\'odka& Pawlow [12].In [2], the expression of any solution to the corresponding stationary problem was obtained.In this paper, assuming $m_{0}=0$ , we shall show that this type of expression of solutions still holds in our non-isothermal setting, even though nonlinear term $g(w)-\lambda'(w)u$ is of general $N$ -shape in $w$ .Furthermore, the structure of the solution set of $P(\sigma_{*}, \sigma^{*}, u_{\infty}, 0)$ and the $\omega$ -limit set of $w$ will be more precisely studied.This paper gives not only some generalizations but also improvements of results [2] to the non-isothermal case.We refer to [8,15,16,17] for related works to the Cahn-Hilliard equation without constraints, and to [3,4,7] for the phase field model with constraint.