INSTABILITY OF THE STANDING WAVES FOR THE NONLINEAR KLEIN-GORDON EQUATIONS IN ONE DIMENSION

In this paper, we consider the nonlinear Klein-Gordon equation Attu-Delta u + u = |u|p-1u, t is an element of R, x is an element of Rd, with 1 < p < 1+ 4d. The equation has the standing wave solutions u omega = ei omega t phi omega with the frequency omega is an element of (-1, 1), where phi omega is the solution of -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 [Comm. Math. Phys. 100 (1985), pp. 173-190] that there exists a critical frequency omega c is an element of (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. Furthermore, the strong instability for the critical frequency |omega| = omega c in the high dimensions d >= 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 |omega| = omega c, p > 1, and d = 1, in which case we prove that the standing wave solution u omega is orbitally unstable.