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Clustering layers and boundary layers in spatially inhomogeneous phase transition problems. (Couches limites dans des problèmes de transition de phase spatialement inhomogènes.) (English) Zbl 1114.35005

Summary: The authors study the existence of solutions with multiple clustered transition layers for the following inhomogeneous problem of Allen-Cahn type: \[ -\varepsilon^2u_{xx}+W_u(x,u)=0 \text{ in } (0,1),\quad u_x(0)=u_x(1)=0,\tag{1} \] where \(\varepsilon>0\) is a small parameter and \(W(x,u)\) is a double-well potential. A typical example of \(W(x,u)\) is \({1\over4}h(x)^2(u^2-1)^2\). In particular, they show the existence of solutions with clustered layers and layers.

MSC:

35B25 Singular perturbations in context of PDEs
35Q35 PDEs in connection with fluid mechanics
34B15 Nonlinear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
47J30 Variational methods involving nonlinear operators
76T99 Multiphase and multicomponent flows
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