Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Structure of Lie Algebras and the Classification Problem for Partial
  Differential Equations   The present paper solves completely the problem of the group classification
of nonlinear heat-conductivity equations of the form\
$u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular,
that the above class contains no nonlinear equations whose invariance algebra
has dimension more than five. Furthermore, we have proved that there are two,
thirty-four, thirty-five, and six inequivalent equations admitting one-, two-,
three-, four- and five-dimensional Lie algebras, respectively. Since the
procedure which we use, relies heavily upon the theory of abstract Lie algebras
of low dimension, we give a detailed account of the necessary facts. This
material is dispersed in the literature and is not fully available in English.
After this algebraic part we give a detailed description of the method and then
we derive the forms of inequivalent invariant evolution equations, and compute
the corresponding maximal symmetry algebras. The list of invariant equations
obtained in this way contains (up to a local change of variables) all the
previously-known invariant evolution equations belonging to the class of
partial differential equations under study.

13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org