×

Unexpectedly linear behavior for the Cahn-Hilliard equation. (English) Zbl 0969.35074

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)].

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
PDFBibTeX XMLCite
Full Text: DOI