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Slow motion for the Cahn-Hilliard equation in one space dimension. (English) Zbl 0753.35042
The authors consider the Cahn-Hilliard equation in one space dimension \[ u_ t=(-\varepsilon^ 2 u_{xx}+W'(u))_{xx}\qquad \text{on}\qquad (0,1)\times(0,\infty),\tag{\text{CH}} \]
\[ u_ x=u_{xx}=0,\quad x=0,1,\quad t>0;\qquad u(x,0)=u_ 0(x). \] Hypothesis: \(W(u)\) is a \(C^ 4\) function with exactly three critical points, \(\alpha<\beta<\gamma\), with \(\alpha,\beta\) local minima and \(\gamma\) a local maximum and \(W\geq 0\), \(W''(\alpha)\), \(W''(\beta)\), \(-W''(\gamma)>0\).
The authors seek the invariant manifold \({\mathcal M}\) for (CH) under the above hypothesis, i.e. seek the solution \(u(x,t)=\hat u(\xi(t),x)\), where \(\hat u(\xi,x)\) and \(\xi(t)\) satisfy \(\dot\xi(t)=b(\xi)\) and \(- \varepsilon^ 2\hat u_{xxxx}+(W'(\hat u_ x))_ x=b(\xi)\hat u\).
They prove the Theorem: There exists \(\varepsilon_ 0>0\) such that for \(\varepsilon<\varepsilon_ 0\) there are functions \(\hat u(\xi,\cdot)\in H^{4\alpha}\) and \(b(\xi)\) defined in \((-\ell+\delta,\ell-\delta)\), with the following properties:
(i) \(\xi\to\hat u(\xi,\cdot)\), \(\xi\to b(\xi)\) are Lipschitz continuous, (ii) \({\mathcal M}=\{u=\hat u(\xi,\cdot);\;\xi\in(-\ell+\delta,\ell- \delta)\}\) is an invariant manifold for the dynamical system defined by (CH) in \(H^{4\alpha}\), (iii) The flow on \({\mathcal M}\) is described by the ordinary differential equation \(\dot\xi=b(\xi)\). (iv) The following estimates hold: \(\| \hat u-u(\xi,\cdot)\|_{H^ 2}=O(\exp(- \nu\delta_ \xi/\varepsilon))\) and \(b(\xi)=O(\exp(-2\nu\delta_ \xi/\varepsilon))\) and, \(b(0)=0\), where \(u(\xi,x)=u(x-\xi)\), \(\delta_ \xi=\ell-|\xi|\), and \(u(x)\) is the equilibrium (time independent solution) of (CH).

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
58J70 Invariance and symmetry properties for PDEs on manifolds
37C10 Dynamics induced by flows and semiflows
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