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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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References:
 [1] DOI: 10.1016/0022-0396(91)90163-4 · Zbl 0753.35042 [2] DOI: 10.1512/iumj.1996.45.1123 · Zbl 0893.35051 [3] DOI: 10.1137/0153049 · Zbl 0788.35061 [4] DOI: 10.1080/03605309608821223 · Zbl 0863.35009 [5] DOI: 10.1016/S0167-2789(96)00190-X · Zbl 0890.35043 [6] Gilbarg D., Elliptic partial differential equations of second order (1977) · Zbl 0361.35003 [7] DOI: 10.1080/03605309308820937 · Zbl 0788.35132 [8] Grinfeld M., Proc. Roy. Soc. Edinburgh Sect. A 125 pp 351– (1995) [9] Grinfeld M., Morse decomposition and structure of the global attractor · Zbl 0927.35045 [10] J.K.Hale, ”Asymptotic behavior of dissipative systems”. (Mathematical Surveys and Monographs 25) Providence: American Mathematical Society 1988 [11] Henry D., Lecture Notes in Mathematics 840 (1981) [12] S.Lojasiewicz, Ensemble semi-analytic. Bures-sur-Yvette: IHES (1965) [13] Lojasiewicz S., Colloque Internationaux du C.N.R.S. 117 pp 87– (1963) [14] Lojasiewicz S., Ann. Inst. Fourier (Grenoble) 43 pp 1575– (1993) · Zbl 0803.32002 [15] Nirenberg L., Topics in Nonlinear Functional Analysis (1974) · Zbl 0286.47037 [16] DOI: 10.1006/jmaa.1998.6066 · Zbl 0919.35022 [17] DOI: 10.2307/2006981 · Zbl 0549.35071 [18] Wei J., Ann. Inst. H. Poincare Anal. Non Lineaire [19] DOI: 10.1006/jdeq.1998.3479 · Zbl 0965.35070 [20] Wei J., SIAM J. Math. Anal. [21] DOI: 10.1080/00036818608839639 · Zbl 0582.34070
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