Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two problems in Partial Differential Equations (in Portuguese)   In this work, we examine two important problems in the theory of nonlinear
PDEs. In Part I, we propose and solve a more general and complete version of
the celebrated Leray's problem for the incompressible Navier-Stokes equations
in $ \mathbb{R}^{3} \!$, which in its simplest form was suggested by
J.$\;$Leray in 1934 (and solved only in the 1980s by T.$\;$ Kato, K.$\;$Masuda
and other authors). A number of related new results of clear interest to the
theory of Leray's solutions are also given here.
  In Part II, which is independent of Part I and can be read separately, we
introduce an important new collection of problems concerning global existence
results and blow-up phenomena for solutions of conservative advection-diffusion
equations in $ \mathbb{R}^{n} $ where heterogeneities in the lower order terms
tend to destabilize the solution (everywhere or in certain regions), strongly
competing with the viscous dissipation effects to determine the overall
solution behavior. Here, we consider the case of superlinear advection (and
arbitrary dimension), which may cause finite-time blow-up in several important
spaces. We then point out a new kind of phenomena --- one that may be properly
named "anti-Fujita" for its vivid contrast to the type of blow-up behavior
discovered by Fujita in the 1960s, and which has been investigated ever since
--- that has apparently been completely overlooked in the literature.

19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph