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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: $$-\varepsilon^2u_{xx}+W_u(x,u)=0 \text{ in } (0,1),\quad u_x(0)=u_x(1)=0,\tag1$$ 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 (PDE) 35Q35 PDEs in connection with fluid mechanics 34B15 Nonlinear boundary value problems for ODE 34E15 Asymptotic singular perturbations, general theory (ODE) 47J30 Variational methods (nonlinear operator equations) 76T99 Two-phase and multiphase flows
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