Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gradient estimates for a nonlinear parabolic equation and Liouville
  theorems   We establish local elliptic and parabolic gradient estimates for positive
smooth solutions to a nonlinear parabolic equation on a smooth metric measure
space. As applications, we determine various conditions on the equation's
coefficients and the growth of solutions that guarantee the nonexistence of
nontrivial positive smooth solutions to many special cases of the nonlinear
equation. In particular, we apply gradient estimates to discuss some
Yamabe-type problems of complete Riemannian manifolds and smooth metric measure
spaces.

9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con