Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations
Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007) no. 4, p. 539-548
@article{AIHPC_2007__24_4_539_0,
     author = {Liu, Yue and Ohta, Masahito and Todorova, Grozdena},
     title = {Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {24},
     number = {4},
     year = {2007},
     pages = {539-548},
     doi = {10.1016/j.anihpc.2006.03.005},
     zbl = {1120.35013},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2007__24_4_539_0}
}
Liu, Yue; Ohta, Masahito; Todorova, Grozdena. Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations. Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007) no. 4, pp. 539-548. doi : 10.1016/j.anihpc.2006.03.005. http://www.numdam.org/item/AIHPC_2007__24_4_539_0/

[1] Berestycki H., Cazenave T., Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris 293 (1981) 489-492. | Zbl 0492.35010

[2] Berestycki H., Gallouët T., Kavian O., Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris 297 (1983) 307-310. | MR 734575 | Zbl 0544.35042

[3] Berestycki H., Lions P.L., Nonlinear scalar field equations, Arch. Rat. Mech. Anal. 82 (1983) 313-345. | MR 695535 | Zbl 0533.35029

[4] Bona J., Sachs R., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988) 15-29. | MR 954673 | Zbl 0654.35018

[5] Boussinesq J., Théorie des ondes et de remous qui se propagent…, J. Math. Pures Appl. 17 (1872) 55-108. | JFM 04.0493.04

[6] Brezis H., Lieb E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486-490. | MR 699419 | Zbl 0526.46037

[7] R.E. Caflisch, Shallow water waves, Lecture notes, New York University, New York.

[8] Cazenave T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI, 2003. | MR 2002047 | Zbl 1055.35003

[9] Cazenave T., Lions P.L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) 549-561. | MR 677997 | Zbl 0513.35007

[10] Fröhlich J., Lieb E.H., Loss M., Stability of Coulomb systems with magnetic fields I. The one-electron atom, Comm. Math. Phys. 104 (1986) 251-270. | MR 836003 | Zbl 0595.35098

[11] Ginibre J., Velo G., The global Cauchy problem for the non linear Klein-Gordon equation, Math. Z. 189 (1985) 487-505. | Zbl 0549.35108

[12] Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987) 160-197. | MR 901236 | Zbl 0656.35122

[13] Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal. 94 (1990) 308-348. | MR 1081647 | Zbl 0711.58013

[14] Kwong M.K., Uniqueness of positive solutions of Δu-u+u p =0 in R n , Arch. Rational Mech. Anal. 105 (1989) 234-266. | MR 969899 | Zbl 0676.35032

[15] Levine H.A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt =-Au+Fu, Trans. Amer. Math. Soc. 192 (1974) 1-21. | MR 344697 | Zbl 0288.35003

[16] Lieb E., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983) 441-448. | MR 724014 | Zbl 0538.35058

[17] Liu Y., Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations 5 (1993) 537-558. | MR 1235042 | Zbl 0784.34048

[18] Liu Y., Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal. 26 (1995) 1527-1546. | MR 1356458 | Zbl 0857.35103

[19] Liu Y., Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations 164 (2000) 223-239. | MR 1765555 | Zbl 0973.35163

[20] Liu Y., Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc. 353 (2000) 191-208. | Zbl 0949.35120

[21] Liu Y., Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions, J. Differential Equations 180 (2002) 153-170. | Zbl 1061.35115

[22] Ohta M., Todorova G., Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dynam. Syst. 12 (2005) 315-322. | Zbl 1065.35198

[23] M. Ohta, G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equation and Klein-Gordon-Zakharov system, Preprint. | Zbl 1065.35198

[24] Payne L.E., Sattinger D.H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975) 273-303. | MR 402291 | Zbl 0317.35059

[25] Shatah J., Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983) 313-327. | Zbl 0539.35067

[26] Shatah J., Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985) 701-710. | Zbl 0617.35072

[27] Shatah J., Strauss W., Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985) 173-190. | MR 804458 | Zbl 0603.35007

[28] Strauss W., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977) 149-162. | MR 454365 | Zbl 0356.35028

[29] Weinstein M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983) 567-576. | MR 691044 | Zbl 0527.35023

[30] Weinstein M.I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986) 51-68. | MR 820338 | Zbl 0594.35005