  The generalized Allen-Cahn equation,
  \[
  u_t=\varepsilon^2(D(u)u_x)_x-\frac{\varepsilon^2}2D'(u)u_x^2-F'(u),
  \]
  with nonlinear diffusion, $D = D(u)$, and potential, $F = F(u)$, of the form
  \[
  D(u) = |1-u^2|^{m}, \quad \text{or} \quad D(u) = |1-u|^{m}, \quad m >1,
  \]
  and
  \[
  F(u)=\frac{1}{2n}|1-u^2|^{n}, \qquad n\geq2,
  \]
  respectively, is studied. These choices correspond to a reaction function
that can be derived from a double well potential, and to a generalized
degenerate diffusivity coefficient depending on the density $u$ that vanishes
at one or at the two wells, $u = \pm 1$. It is shown that interface layer
solutions that are equal to $\pm 1$ except at a finite number of thin
transitions of width $\varepsilon$ persist for an either exponentially or
algebraically long time, depending upon the interplay between the exponents $n$
and $m$. For that purpose, energy bounds for a renormalized effective energy
potential of Ginzburg-Landau type are derived.
