# American Institute of Mathematical Sciences

• Previous Article
Boundary control problems with convex cost and dynamic programming in infinite dimension part II: Existence for HJB
• DCDS Home
• This Issue
• Next Article
Homogenization of second order equation with spatial dependent coefficient
February  2005, 12(2): 315-322. doi: 10.3934/dcds.2005.12.315

## Strong instability of standing waves for nonlinear Klein-Gordon equations

 1 Department of Mathematics, Faculty of Science, Saitama University, Japan 2 Department of Mathematics, University of Tennessee, Knoxville, TN 37096-1300, United States

Received  July 2003 Revised  June 2004 Published  December 2004

The strong instability of ground state standing wave solutions $e^{i\omega t}\phi_{\omega}(x)$ for nonlinear Klein-Gordon equations has been known only for the case $\omega=0$. In this paper we prove the strong instability for small frequency $\omega$.
Citation: Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315
 [1] Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 [2] Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 [3] Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389 [4] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [5] José R. Quintero, Juan C. Cordero. Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1213-1240. doi: 10.3934/dcdsb.2019217 [6] Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525 [7] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [8] Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215 [9] Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233 [10] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 [11] Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067 [12] Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure and Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413 [13] Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043 [14] Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy. Electronic Research Archive, 2020, 28 (2) : 671-689. doi: 10.3934/era.2020035 [15] Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889 [16] François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229 [17] Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903 [18] Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076 [19] Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679 [20] Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251

2020 Impact Factor: 1.392