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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.

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
Full Text: DOI
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