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Long-time behaviour for a model of phase-field type. (English) Zbl 0851.35055
The long-time behaviour of a model of phase-field type is investigated in this paper. This model consists of a coupled system of two parabolic equations: $\varphi_t- k_1 \Delta\varphi= - s'(\varphi)+ w'(\varphi) u,\quad u_t- k_2 \Delta u= - w'(\varphi) \varphi_t\quad \text{in } Q,\tag{1}$ ${\partial\varphi\over \partial n}= 0,\quad {\partial u\over \partial n}= 0\quad \text{on }\Sigma,\quad\varphi(\cdot, 0)= \varphi_0,\quad u(\cdot, 0)= u_0\quad \text{in } \Omega,$ where $$\Omega$$ is a bounded open subset of $$\mathbb{R}^N$$, $$1\leq N\leq 3$$, with smooth boundary $$\Gamma$$, $$Q= \Omega\times (0, +\infty)$$, $$\Sigma= \Gamma\times (0, +\infty)$$, $$k_1$$, $$k_2$$ are positive real numbers and functions $$w, s\in C^\infty(\mathbb{R})$$ satisfy suitable conditions. It describes the time evolution of a binary system which occupies a region $$\Omega$$ and whose state may be described by an order parameter $$\varphi$$ and by the temperature $$u$$.
The author reformulates the problem (1) using the order parameter $$\varphi$$ and the energy density $$e= u+ w(\varphi)$$, it gives: $\varphi_t- k_1 \Delta\varphi= - s'(\varphi)- w'(\varphi) w(\varphi)+ w'(\varphi) e, e_t- k_2 \Delta e= - k_2 \text{ div}(w'(\varphi) \nabla \varphi)\text{ in } Q,\tag{2}$ ${\partial \varphi\over \partial n}= 0,\quad {\partial e\over \partial n}= 0\quad \text{on }\Sigma,\quad \varphi(\cdot, 0)= \varphi_0,\quad e(\cdot, 0)= u_0+ w(\varphi_0)= e_0\quad \text{in }\Omega.$ There are introduced the following function spaces $X_\beta= \Biggl\{ (\varphi, e)\in W^{1, 4} (\Omega, \mathbb{R}^2): \int_\Omega e(x) dx= |\Omega|\beta\Biggr\},\quad Y_\alpha= \bigcup_{|\beta|\leq \alpha} X_\beta,$ for any real number $$\beta$$ and for any nonnegative real number $$\alpha$$.
Using abstract results of Amann, the well-posedness of (2) for initial data $$(\varphi_0, e_0)$$ in $$W^{1,4}(\Omega, \mathbb{R}^2$$) is proved. Moreover, there is proved that $$\{S_t\} (S_t(\varphi_0, e_0)= (\varphi(t), e(t)))$$ is a strongly continuous semigroup in $$W^{1, 4}(\Omega, \mathbb{R}^2)$$ and maps $$X_\beta$$ in itself for each real number $$\beta$$. Finally, there is proved that the semigroup $$\{S_t\}$$ has a maximal attractor in $$Y_\alpha$$ for any nonnegative real number $$\alpha$$ and an exponential attractor in $$L^2(\Omega)\times H^1(\Omega)'$$.

##### MSC:
 35K45 Initial value problems for second-order parabolic systems 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations
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##### References:
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