Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A curve shortening equation with time-dependent mobility related to
  grain boundary motions   A curve shortening equation related to the evolution of grain boundaries is
presented. This equation is derived from the grain boundary energy by applying
the maximum dissipation principle. Gradient estimates and large time asymptotic
behavior of solutions are considered. In the proof of these results, one key
ingredient is a new weighted monotonicity formula that incorporates a
time-dependent mobility.

25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph