Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension

In this paper, we consider the nonlinear Klein-Gordon equation ∂ t t u − Δ u + u = | u | p − 1 u , t ∈ R , x ∈ R d , \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 + 4 d 1>p> 1+\frac {4}{d} . The equation has the standing wave solutions u ω = e i ω t ϕ ω u_\omega =e^{i\omega t}\phi _{\omega } with the frequency ω ∈ ( − 1 , 1 ) \omega \in (-1,1) , where ϕ ω \phi _{\omega } is the solution of − Δ ϕ + ( 1 − ω 2 ) ϕ − ϕ p = 0. \begin{align*} -\Delta \phi +(1-\omega ^2)\phi -\phi ^p=0. \end{align*} It was proved by Shatah [Comm. Math. Phys. 91 (1983), pp. 313–327], and Shatah-Strauss [Comm. Math. Phys. 100 (1985), pp. 173–190] that there exists a critical frequency ω c ∈ ( 0 , 1 ) \omega _c\in (0,1) such that the standing waves solution u ω u_\omega is orbitally stable when ω c > | ω | > 1 \omega _c>|\omega |>1 , and orbitally unstable when | ω | > ω c |\omega |>\omega _c . Furthermore, the strong instability for the critical frequency | ω | = ω c |\omega |=\omega _c in the high dimensions d ≥ 2 d\ge 2 was proved by Ohta-Todorova [SIAM J. Math. Anal. 38 (2007), pp. 1912–1931]. In this paper, we settle the only remaining problem when | ω | = ω c |\omega |=\omega _c , p > 1 p> 1 , and d = 1 d=1 , in which case we prove that the standing wave solution u ω u_\omega is orbitally unstable.