Boundedness of semilinear Duffing equations with singularity

Abstract

We prove the boundedness of all solutions for the equation x″ + V′(x) = D_xG(x, t), where V(x) is of singular potential, i.e., lim_x→-1V(x) = +∞, and G(x, t) is bounded and periodic in t. We give sufficient conditions on V(x) and G(x, t) to ensure that all solutions are bounded.

Keywords

Hamiltonian system / repulsive singularity / boundedness of solutions / canonical transformation / Moser’s small twist theorem

References

[1] Alonso J, Ortega R. Unbounded solutions of semilinear equations at resonance. Nonlinearity, 1996, 9: 1099-1111

[2] Arnold V. On the behavior of an adiabatic invariant under a slow periodic change of the Hamiltonian. Dokl Akad Nauk, 1962, 142(4): 758-761 (Transl Sov Math Dokl, 3: 136-139)

[3] Capietto A, Dambrosio W, Liu B. On the boundedness of solutions to a nonlinear singular oscillator. Z Angew Math Phys, 2009, 60: 1007-1034

[4] Dieckerhoff R, Zehnder E. Boundedness of solutions via the twist theorem. Ann Sc Norm Super Pisa Cl Sci, 1987, 14: 79-95

[5] Jiao L, Piao D, Wang Y. Boundedness for general semilinear Duffing equations via the twist theorem. J Differential Equations, 2012, 252: 91-113

[6] Lazer A, Leach D. Bounded perturbations of forced harmonic oscillators at resonance. Ann Mat Pura Appl, 1969, 82: 49-68

[7] Levi M. Quasiperiodic motions in superquadratic time-periodic potentials. Comm Math Phys, 1991, 143: 43-83

[8] Littlewood J. Unbounded solutions of y″ + g(y) = p(t). J Lond Math Soc, 1966, 41: 491-496

[9] Liu B. Boundedness in nonlinear oscillations at resonance. J Differential Equations, 1999, 153: 142-174

[10] Liu B. Boundedness in asymmetric oscillations. J Math Anal Appl, 1999, 231: 355-373

[11] Liu B. Quasi-periodic solutions of forced isochronous oscillators at resonance. J Differential Equa<?Pub Caret?>tions, 2009, 246: 3471-3495

[12] Ma S, Wu J. A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance. Nonlinear Anal, 2007, 67(1): 200-237

[13] Mawhin J. Resonance and nonlinearity: A survey. Ukrainian Math J, 2007, 59(2): 197-214

[14] Morris G. A case of boundedness in Littlewood’s problem on oscillatory differential equations. Bull Aust Math Soc, 1976, 14: 71-93

[15] Moser J. On invariant curves of area-preserving mappings of an annulus. Nachr Akad Wiss Göttingen Math-Phys Kl II, 1962, 1: 1-20

[16] Ortega R. Boundedness in a piecewise linear oscillator and a variant of the small twist theorem. Proc Lond Math Soc, 1999, 79: 381-413

[17] Xu J, You J. Persistence of lower-dimensional tori under the first Melnikov’s nonresonance condition. J Math Pures Appl, 2001, 80(10): 1045-1067

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