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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:
35B25Singular perturbations (PDE)
35Q35PDEs in connection with fluid mechanics
34B15Nonlinear boundary value problems for ODE
34E15Asymptotic singular perturbations, general theory (ODE)
47J30Variational methods (nonlinear operator equations)
76T99Two-phase and multiphase flows
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