Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity

This paper considers a one-dimensional generalized Allen–Cahn equation of the form $$\begin{aligned} u_t = \varepsilon ^2 (D(u)u_x)_x - f(u), \end{aligned}$$ 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.