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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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