Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the transition from parabolicity to hyperbolicity for a nonlinear
  equation under Neumann boundary conditions   An integro differential equation which is able to describe the evolution of a
large class of dissipative models, is considered. By means of an equivalence,
the focus shifts to the perturbed sine- Gordon equation that in
superconductivity finds interesting applications in multiple engineering areas.
The Neumann boundary problem is considered, and the behaviour of a viscous
term, defined by a high order derivative with small diffusion coefficient , is
investigated. The Green function, expressed by means of Fourier series, is
considered, and an estimate is achieved. Furthermore, some classes of solutions
of the hyperbolic equation are determined, proving that there exists at least
one solution with bounded derivatives. Results obtained prove that diffusion
effects are bounded and tend to zero when e tends to zero.

10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex