Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Instability of the standing waves for the nonlinear Klein-Gordon
  equations in one dimension   In this paper, we consider the following nonlinear Klein-Gordon equation
\begin{align*} \partial_{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb{R},\
x\in \mathbb{R}^d, \end{align*} with $1<p< 1+\frac{4}{d}$. The equation has the
standing wave solutions $u_\omega=e^{i\omega t}\phi_{\omega}$ with the
frequency $\omega\in(-1,1)$, where $\phi_{\omega}$ obeys \begin{align*} -\Delta
\phi+(1-\omega^2)\phi-\phi^p=0. \end{align*} It was proved by Shatah (1983),
and Shatah, Strauss (1985) that there exists a critical frequency $\omega_c\in
(0,1)$ such that the standing waves solution $u_\omega$ is orbitally stable
when $\omega_c<|\omega|<1$, and orbitally unstable when $|\omega|<\omega_c$.
Further, the critical case $|\omega|=\omega_c$ in the high dimension $d\ge 2$
was considered by Ohta, Todorova (2007), who proved that it is strongly
unstable, by using the virial identities and the radial Sobolev inequality. The
one dimension problem was left after then. In this paper, we consider the
one-dimension problem and prove that it is orbitally unstable when
$|\omega|=\omega_c$.

15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph