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Multi-clustered high-energy solutions for a phase transition problem. (English) Zbl 1148.34319

Summary: We study the balanced Allen-Cahn problem in a singular perturbation setting \[ \begin{gathered} -\varepsilon^2 \frac{d^2u}{dx^2}+ a(x)W'(u)=0,\quad x\in (0,1),\\ \left.\frac{du}{dx}\right|_{x=0}=\left.\frac{du}{dx}\right|_{x=1}=0.\end{gathered} \] . We are interested in the behaviour of clusters of layers, i.e. a family of solutions \(u_{\varepsilon}(x)\) with an increasing number of layers as \(\varepsilon\to 0\). In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

MSC:

34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
74N99 Phase transformations in solids
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