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Zbl 1173.35028
Song, Lingyu; Zhang, Yindi; Ma, Tian
Global attractor of the Cahn-Hilliard equation in $H^k$ spaces. (English)
[J] J. Math. Anal. Appl. 355, No. 1, 53-62 (2009). ISSN 0022-247X

It is well-known that the Cahn-Hilliard equation $$ \cases \partial_t u = -\Delta\left( \Delta u - g(u) \right), &(x,t)\in \Omega\times (0,\infty),\\ \partial_n u = \partial_n\Delta u = 0, &(x,t)\in \partial\Omega\times (0,\infty), \endcases$$ has a compact attractor $\mathcal{A}$ in the space $L^2_0(\Omega)$ of square integrable functions with zero mean-value when $\Omega$ is a bounded open subset of ${\mathbb R}^n$, $1\le n \le 3$, and $g(s)=\sum_{k=1}^p a_k s^k$ with $a_p>0$, $p\ge 3$ an odd integer if $n=1,2$, and $p=3$ if $n=3$. Using an iteration procedure relying on parabolic regularizing effects, the attractor $\mathcal{A}$ is shown to be compact in $H^k(\Omega)$ for all $k\ge 1$ and to attract the dynamics in $H^k(\Omega)$ as well.
[Philippe Laurençot (Toulouse)]

MSC 2000:: *35B41 Attractors
80A22 Stefan problems, etc.
35K55 Nonlinear parabolic equations
35K35 Higher order parabolic equations, boundary value problems

Keywords: compact attractor; iteration procedure

Cited in: Zbl 1242.35058

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