Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity. (English) Zbl 1445.35025
Summary: This paper considers a one-dimensional generalized Allen-Cahn equation of the form
\[
u_t = \varepsilon^2 (D(u)u_x)_x - f(u),
\]
where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field \(u\), and \(f(u)\) is a reaction function that can be derived from a double-well potential with minima at two pure phases \(u = \alpha\) and \(u = \beta\). It is shown that interface layers (namely, solutions that are equal to \(\alpha\) or \(\beta\) except at a finite number of thin transitions of width \(\varepsilon)\) persist for an exponentially long time proportional to \(\exp (C/\varepsilon)\), where \(C > 0\) is a constant. In other words, the emergence and persistence of metastable patterns for this class of equations is established. For that purpose, we prove energy bounds for a renormalized effective energy potential of Ginzburg-Landau type. Numerical simulations, which confirm the analytical results, are also provided.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K59 | Quasilinear parabolic equations |
| 35B36 | Pattern formations in context of PDEs |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
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