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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