Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Supersolutions for parabolic equations with unbounded diffusion and its
  applications to some classes of parabolic and hyperbolic equations   This paper is concerned with supersolutions to parabolic equations of the
form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in
\mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is
positive. Under the behavior of the diffusion coefficient $D$ with polynomial
order at spatial infinity, a family of supersolutions with slowly decaying
property at spatial infinity is provided. As a first application, weighted
$L^2$ type decay estimates for the initial-boundary value problem of the
corresponding parabolic equation are proved. The second application is the
study of the exterior problem of wave equations with space-dependent damping
terms. By using supersolutions provided above, energy estimates with polynomial
weight and diffusion phenomena are shown.

29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th