Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long-time behavior for the porous medium equation with small initial
  energy   We study the long-time behavior for the solution of the Porous Medium
Equation in an open bounded connected set, with smooth boundary. Homogeneous
Dirichlet boundary conditions are considered. We prove that if the initial
datum has sufficiently small energy, then the solution converges to a
nontrivial constant sign solution of a sublinear Lane-Emden equation, once
suitably rescaled. We point out that the initial datum is allowed to be
sign-changing. We also give a sufficient energetic criterion on the initial
datum, which permits to decide whether convergence takes place towards the
positive solution or to the negative one.

32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph