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Convergence of solutions to Cahn-Hilliard equation. (English) Zbl 0936.35032
The authors show the convergence to an equilibrium as \(t \to \infty\) of solutions to the Cahn-Hilliard equation \[ u_t= \Delta ( -{\varepsilon}^2 \Delta u + W_u (u)), \;u(0)=u_0\text{ in }\Omega; \quad {{\partial u} \over {\partial \nu}}=0={{\partial} \over {\partial \nu}} (-{\varepsilon}^2 \Delta u + W_u (u))\text{ on }\partial\Omega. \] \(W\) is assumed to be analytic in \(u,\) which allows them to proof a Łojasiewicz inequality for the associated gradient flow in \(W^{-1,2} (\Omega),\) which they use in turn to prove the convergence for the associated class of nonlinear parabolic equations by a method modeled by L. Simon [Ann. Math., II. Ser. 118, 525-571 (1983; Zbl 0549.35071)]. The paper also contains an existence and uniqueness result with the appropriate regularity for solutions providing the background for the study of the asymptotic behaviour.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
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