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 u − mathvariant="normal">Δ<!-- Δ <mml:mo>+ = | p 1 , ∈<!-- ∈ mathvariant="double-struck">R x d \begin{align*} \partial _{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb {R},\ x\in {R}^d, \end{align*} with &gt; 4 encoding="application/x-tex">1&gt;p&gt; 1+\frac {4}{d} . The has standing wave solutions alttext="u omega e i phi omega"> ω<!-- ω <mml:mi>e i ϕ<!-- ϕ encoding="application/x-tex">u_\omega =e^{i\omega t}\phi _{\omega } frequency alttext="omega left-parenthesis negative right-parenthesis"> stretchy="false">( stretchy="false">) encoding="application/x-tex">\omega \in (-1,1) , where alttext="phi encoding="application/x-tex">\phi is solution of squared right-parenthesis 0 period 2 0. -\Delta \phi +(1-\omega ^2)\phi -\phi ^p=0. It was proved by Shatah [Comm. Math. Phys. 91 (1983), pp. 313–327], and Shatah-Strauss 100 (1985), 173–190] that there exists a critical c c 0 _c\in (0,1) such waves encoding="application/x-tex">u_\omega orbitally stable when 1"> _c&gt;|\omega |&gt;1 unstable alttext="StartAbsoluteValue c"> encoding="application/x-tex">|\omega |&gt;\omega _c Furthermore, strong instability for |=\omega in high dimensions alttext="d greater-than-or-equal-to 2"> ≥<!-- ≥ encoding="application/x-tex">d\ge 2 Ohta-Todorova [SIAM J. Anal. 38 (2007), 1912–1931]. settle only remaining problem alttext="p encoding="application/x-tex">p&gt; 1 encoding="application/x-tex">d=1 which case prove unstable.