Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long-time behavior of solutions to the generalized Allen-Cahn model with
  degenerate diffusivity   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.

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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: econ
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph