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On the slow dynamics for the Cahn-Hilliard equation in one space dimension. (English) Zbl 0777.35007
The following one-dimensional Cahn-Hilliard equation is considered \[ u_ t=(-\varepsilon^ 2 u_{xx}+2(u^ 3-u))_{xx},\quad\text{in }(a,b)\times\mathbb{R}^ +\tag{1} \] with initial condition (2) \(u(x,0)=u^ \varepsilon_ 0(x)\), \(x\in (a,b)\), and boundary conditions of the type (3) \(u_ x(a,t)=u_ x(b,t)=u_{xxx}(a,t)=u_{xxx}(b,t)=0\), \(t>0\). It is assumed that the spinodal decomposition has occurred, i.e. \(u^ \varepsilon_ 0(x)=\pm 1\) except near a finite number of transition points. The paper analyzes the evolution of the transition layers as \(\varepsilon\) tends to zero; using energy-type estimates, it is proved that the transition points move slower then any power of \(\varepsilon\). With respect to previous results of the same type, the analysis (although not giving a precise estimate of the velocity) has the advantage that the arguments used here are simpler and that weaker assumptions on the initial datum are made (in particular any finite number of transition layers can be handled).
Moreover, the results also apply to the case of Dirichlet boundary conditions \(u(a,t)=\pm 1\), \(u(b,t)=\pm 1\), \(u_{xx}(a,t)=u_{xx}(b,t)=0\), \(t>0\).

MSC:
35B25 Singular perturbations in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
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