Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A gradient flow generated by a nonlocal model of a neural field in an
  unbounded domain   In this paper we consider the non local evolution equation $$ \frac{\partial
u(x,t)}{\partial t} + u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)dy+
h(x). %\,\,\, h \geq 0. $$ We show that this equation defines a continuous flow
in both the space $C_{b}(\mathbb{R}^{N})$ of bounded continuous functions and
the space $C_{\rho}(\mathbb{R}^{N})$ of continuous functions $u$ such that $u
\cdot \rho$ is bounded, where $\rho $ is a convenient "weight function"'. We
show the existence of an absorbing ball for the flow in $C_{b}(\mathbb{R}^{N})$
and the existence of a global compact attractor for the flow in
$C_{\rho}(\mathbb{R}^{N})$, under additional conditions on the nonlinearity. We
then exhibit a continuous Lyapunov function which is well defined in the whole
phase space and continuous in the $C_{\rho}(\mathbb{R}^{N})$ topology, allowing
the characterization of the attractor as the unstable set of the equilibrium
point set. We also illustrate our result with a concrete example.

27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph