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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.

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
Full Text: DOI Numdam EuDML
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