Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global relaxation of bistable solutions for gradient systems in one
  unbounded spatial dimension   This paper is concerned with spatially extended gradient systems of the form
\[ u_t=-\nabla V (u) + \mathcal{D} u_{xx}\,, \] where spatial domain is the
whole real line, state-parameter $u$ is multidimensional, $\mathcal{D}$ denotes
a fixed diffusion matrix, and the potential $V$ is coercive at infinity.
"Bistable" solutions, that is solutions close at both ends of space to stable
homogeneous equilibria, are considered. For a solution of this kind, it is
proved that, if the homogeneous equilibria approached at both ends belong to
the same level set of the potential and if an appropriate (localized in space)
energy remains bounded from below when time increases, then the solution
approaches, when time approaches infinity, a pattern of stationary solutions
homoclinic or heteroclinic to homogeneous equilibria. This result provides a
step towards a complete description of the global behaviour of all bistable
solutions that is pursued in a companion paper. Some consequences are derived,
and applications to some examples are given.

11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga