Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Time-periodic solutions of contact Hamilton-Jacobi equations on the
  circle   We are concerned with the existence and multiplicity of nontrivial
time-periodic viscosity solutions to \[ \partial_t w(x,t) + H( x,\partial_x
w(x,t),w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty). \] We find that
there are infinitely many nontrivial time-periodic viscosity solutions with
different periods when $\frac{\partial H}{\partial u}(x,p,u)\leqslant-\delta<0$
by analyzing the asymptotic behavior of the dynamical system $(C(\mathbb{S}
,\mathbb{R}),\{T_t\}_{t\geqslant 0})$, where $\{T_t\}_{t\geqslant 0}$ was
introduced in \cite{WWY1}. Moreover, in view of the convergence of
$T_{t_n}\varphi$, we get the existence of nontrivial periodic points of $T_t$,
where $\varphi$ are initial data satisfying certain properties. This is a
long-time behavior result for the solution to the above equation with initial
data $\varphi$. At last, as an application, we describe to readers a
bifurcation phenomenon for \[ \partial_t w(x,t) + H( x,\partial_x
w(x,t),\lambda w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty), \] when
the sign of the parameter $\lambda$ varies. The structure of the unit circle
$\mathbb{S}$ plays an essential role here. The most important novelty is the
discovery of the nontrivial recurrence of $(C(\mathbb{S}
,\mathbb{R}),\{T_t\}_{t\geqslant 0})$.

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The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas