Sander, Evelyn; Wanner, Thomas Unexpectedly linear behavior for the Cahn-Hilliard equation. (English) Zbl 0969.35074 SIAM J. Appl. Math. 60, No. 6, 2182-2202 (2000). Let (1) \(u_t= -\Delta(\varepsilon^2\Delta u+ f(u))\) in \(\Omega\) and \({\partial u\over\partial v}={\partial\Delta u\over\partial v}= 0\) on \(\partial\Omega\), be the Cahn-Hilliard equation, where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(n\leq 3\), and \(-f\) is the derivative of a double-well potential \(F\). This paper deals with the spinodal decomposition of binary alloys, i.e. with solutions of (1) originating near the homogeneous equilibrium \(\overline u_0=\mu\), where \(\mu\) belongs to the spinodal interval, i.e. \(f'(\mu)> 0\). The main result of the paper is stated under a geometric condition on \(\Omega\), satisfied, in particular, for rectangular domains, when \(f(u)= u-u^{1+\sigma}\) for some \(\sigma\geq 1\), and \(\mu= 0\). The authors give conditions on the initial condition \(u_0\) which imply that the solution \(u\) of (1), originating at \(u_0\), closely follows the corresponding solution \(v\) of the linearization of (1) at \(\overline u_0\), up to a ball of unexpectedly large radius \(R_\varepsilon\) in the \(H^2(\Omega)\)-norm \(\|\cdot\|\); they also obtain an estimate of the relative distance \(\|u(t)- v(t)\|/\|v(t)\|\). The radius \(R_\varepsilon\) is growing as \(\varepsilon\to 0\), and its size is better than in the results of S. Maier-Paape and T. Wanner [Arch. Ration. Mech. Anal. 151, No. 3, 187-219 (2000; Zbl 0954.35089)]. Their approach is based on properties of abstract evolution equations and, in particular, on results of [D. Henry, Lecture Notes in Mathematics 840, Springer-Verlag (1981; Zbl 0456.35001)]. Reviewer: Denise Huet (Nancy) Cited in 1 ReviewCited in 17 Documents MSC: 35K35 Initial-boundary value problems for higher-order parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs Keywords:spinodal decomposition; phase separation; pattern formation Citations:Zbl 0954.35089; Zbl 0456.35001 PDFBibTeX XMLCite \textit{E. Sander} and \textit{T. Wanner}, SIAM J. Appl. Math. 60, No. 6, 2182--2202 (2000; Zbl 0969.35074) Full Text: DOI