Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the evolution of subcritical regions for the Perona-Malik equation   The Perona-Malik equation is a celebrated example of forward-backward
parabolic equation. The forward behavior takes place in the so-called
subcritical region, in which the gradient of the solution is smaller than a
fixed threshold. In this paper we show that this subcritical region evolves in
a different way in the following three cases: dimension one, radial solutions
in dimension greater than one, general solutions in dimension greater than one.
  In the first case subcritical regions increase, but there is no estimate on
the expansion rate. In the second case they expand with a positive rate and
always spread over the whole domain after a finite time, depending only on the
(outer) radius of the domain. As a by-product, we obtain a non-existence result
for global-in-time classical radial solutions with large enough gradient. In
the third case we show an example where subcritical regions do not expand.
  Our proofs exploit comparison principles for suitable degenerate and
non-smooth free boundary problems.

34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an