Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Schwarz symmetrizations in parabolic equations on complete manifolds   In this article, we prove a sharp estimate for the solutions to parabolic
equations on manifolds. Precisely, using symmetrization techniques and
isoperimetric inequalities on Riemannian manifold, we obtain a Bandle's
comparison on complete noncompact manifolds with nonnegative Ricci curvature
and compact manifolds with positive Ricci curvature respectively. Our results
generalize Bandle's result [6] to Riemannian setting, and Talenti's comparison
for elliptic equation on manifolds by Colladay-Langford-McDonald [12] and
Chen-Li [9] to parabolic equations.

30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin