Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gradient estimates for some $f$-heat equations driven by Lichnerowicz's
  equation on complete smooth metric measure spaces   Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the
Bakry-\'Emery Ricci curvature bounded from below, various gradient estimates
for solutions of the following general $f$-heat equations $$ u_t=\Delta_f
u+au\log u+bu +Au^p+Bu^{-q} $$ and \[ u_t=\Delta_f u+Ae^{pu}+Be^{-pu}+D \] are
studied. As by-product, we obtain some Liouville-type theorems and Harnack-type
inequalities for positive solutions of several nonlinear equations including
the Schr\"{o}dinger equation, the Yamabe equation, and Lichnerowicz-type
equations as special cases.

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The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys