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Finite dimensional dynamics and interfaces intersecting the boundary: Equilibria and quasi-invariant manifold. (English) Zbl 0893.35051
The authors study the following single reaction-diffusion equation (Allen-Cahn type) in a bounded domain for small \(\varepsilon>0\), \[ u_t= \varepsilon^2\Delta u-f(u) \text{ in }\Omega \times(0, \infty), \] with Neumann boundary condition, where the nonlinear term \(f\) is a bistable type with two zeros. The domain \(\Omega\) is assumed to have a rectangular shaped portion. They construct a stable equilibrium solution with a thin flat sharp layer, which interface intersects the flat boundary nearly orthogonally. The very slow dynamics of the solution around this equilibrium solution is also investigated.
Reviewer: S.Jimbo (Sapporo)

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B25 Singular perturbations in context of PDEs
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