Instability of standing waves for Hamiltonian wave equations

Abstract

This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest. On the one hand, by proving a compactness lemma and solving a variational problem, the existence of the standing wave with ground state for the aforementioned equation is proved. On the other hand, the authors derive the instability of the standing wave by applying the potential well argument, the concavity method and an invariant region under the solution flow of the Cauchy problem for the equation under study, and the invariance of the region aforementioned can be shown by introducing an auxiliary functional and a supplementary constrained variational problem.

Keywords

Hamiltonian wave equation / Ground state / Standing wave / Instability

References

[1] Soffer A., Weinstein M. I.. Resonance, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Mat., 1999, 136: 9-74

[2] Pecher H.. L ^p-Abschätzungen und klassiche für nichtlineare Wellengeichungen. I. Math. Z., 1976, 150: 159-183

[3] Levine H. A.. Instability and non-existence of global solutions to nonlinear wave equations of the form Pu _tt = −Au + F(u). Trans. Amer. Math. Soc., 1974, 192: 1-21

[4] Payne L. E., Sattinger D. H.. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math., 1975, 22(3–4): 273-303

[5] Ball J. M.. Grandall M. G.. Finite time blow-up in nonlinear problems, Nonlinear Evolution Equations, 1978, New York: Academic Press 189-205

[6] Strauss W. A.. Nonlinear wave equations, 1989, Providence, RI: A. M. S.

[7] Shatah J.. Stable standing waves of nonlinear Klein-Gordon equations. Comm. Math. Phys., 1983, 91: 313-327

[8] Shatah J.. Unstable ground state of nonlinear Klein-Gordon equations. Trans. Amer. Math. Soc., 1985, 290(2): 701-710

[9] Berestycki H., Lions P. L.. Nonlinear scalar field equations I, II. Arch. Rat. Mech. Anal., 1983, 82: 313-375

[10] Lions P. L.. The concentration compactness principle in the calculus of variations, I, The locally compact case. Ann. Inst. Inst. H. Poincaré, Anal. Non Linéaire, 1984, 1: 109-145

[11] Nirenberg L.. Topics in nonlinear functional analysis, 1974, New York: New York University

[12] Rose H. A., Weinstein M. I.. On the bound states of the nonlinear Schrödinger equation with a linear potential. Phys. D, 1988, 30: 207-218

[13] Grillakis M., Shatah J., Strauss W.. Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal., 1987, 74: 160-197

[14] Weinstein M. I.. Lyapunov stability of ground states of nonlinear dispersive ebolution equations. Comm. Pure Appl. Math., 1986, 39: 51-68

[15] Zhang J.. Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential. Commun. Part. Diff. Eqs., 2005, 30: 1429-1443

[16] Zhang J.. Stability of attractive Bose-Einstein condensates. J. Stat. Phys., 2000, 101: 731-746

[17] Fukuizumi R.. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Disc. Conti. Dyn. Sys., 2001, 7(3): 525-544

[18] Oh Y. G.. Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials. J. Diff. Eqs., 1989, 81: 255-274

[19] Cazenave T.. An Introduction to Nonlinear Schrödinger Equations, 1989, Rio de Janeiro: Instituto de Mathemática-UFRJ