Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the dynamics of a non-local parabolic equation arising from the
  Gierer-Meinhardt system   The purpose of the current paper is to contribute to the comprehension of the
dynamics of the shadow system of an activator-inhibitor system known as a
Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate
step between single equations and reaction-diffusion systems. In the case where
the inhibitor's response to the activator's growth is rather weak, then the
shadow system of the Gierer-Meinhardt model is reduced to a single though
non-local equation whose dynamics will be investigated. We mainly focus on the
derivation of blow-up results for this non-local equation which can be seen as
instability patterns of the shadow system. In particular, a {\it diffusion
driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of
a constant stationary solution, which it is destabilised via diffusion-driven
blow-up, is obtained. The latter actually indicates the formation of some
unstable patterns, whilst some stability results of global-in-time solutions
towards non-constant steady states guarantee the occurrence of some stable
patterns.

16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs