Sharp Threshold for Blowup and Global Existence in Nonlinear Schrödinger Equations Under a Harmonic Potential

Jian, Zhang

doi:10.1080/03605300500299539

Cited by 116 publications

(91 citation statements)

References 30 publications

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“…In this case, the global existence in the energy spaceH 1 has been proved for (1) 1 ≤ p < d+2 (d−2) + without smallness assumption on the Cauchy data in the defocusing case (λ < 0) and for small Cauchy data in the focusing case (λ > 0) (see [Car02] and also [Zha05]). But nothing is known for nonlinearities of higher order, neither about conservation of theH s -norm for s > 1.…”

Section: Introduction Statement Of the Resultsmentioning

confidence: 99%

Normal Forms for Semilinear Quantum Harmonic Oscillators

Grébert

Imekraz

Paturel

2009

Commun. Math. Phys.

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Abstract. -We consider the semilinear harmonic oscillator iψt = (−∆ + |x| 2 + M )ψ + ∂2g(ψ,ψ), x ∈ R d , t ∈ R where M is a Hermite multiplier and g a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on M related to the non resonance of the linear part, this normal form is integrable when d = 1 and gives rise to simple (in particular bounded) dynamics when d ≥ 2. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.Résumé (Formes normales de Birkhoff pour l'oscillateur harmonique quantique non linéaire)Dans cet article nous considérons l'oscillateur harmonique semi-linéaire :

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Section: Introduction Statement Of the Resultsmentioning

confidence: 99%

Normal Forms for Semilinear Quantum Harmonic Oscillators

Grébert

Imekraz

Paturel

2009

Commun. Math. Phys.

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“…For the non-linear Klein-Gordon equation with a non-negative potential, Gan and Zhang [5] obtained a sharp threshold of blowup and global existence for its solution by using the method proposed in [24]. For the study of the non-linear Schrödinger equation with a harmonic potential, Zhang [25] derived a sharp threshold of blowup and global existence for its solution by introducing a type of cross-constrained variational method. Own to the natural relation between the Klein-Gordon equation and the Schrödinger equation, it is interesting to apply the method given in [25] to obtain a sharp threshold of blowup and global existence for the non-linear Klein-Gordon equation with a potential.…”

Section: Introductionmentioning

confidence: 99%

“…For the study of the non-linear Schrödinger equation with a harmonic potential, Zhang [25] derived a sharp threshold of blowup and global existence for its solution by introducing a type of cross-constrained variational method. Own to the natural relation between the Klein-Gordon equation and the Schrödinger equation, it is interesting to apply the method given in [25] to obtain a sharp threshold of blowup and global existence for the non-linear Klein-Gordon equation with a potential. Unfortunately, as far as our knowledge is concerned, the method in [25] cannot apply to the Klein-Gordon equation with a general potential except the inverse square potential |x| −2 .…”

Section: Introductionmentioning

confidence: 99%

Cross-Constrained Variational Problem and the Non-Linear Klein–gordon Equations

Gan¹,

Zhang²

2008

Glasgow Math. J.

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Abstract. In this paper, we put forward a cross-constrained variational method to study the non-linear Klein-Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.2000 Mathematics Subject Classification. 35A15, 35L15.

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“…At the heart of our approach is an additional characterization of the ground states as being at a mountain pass level for S. This characterization was derived in [10] for N 2 and in [11] for N = 1. We also strongly benefit from recent techniques developed by several authors [12,13,14,15,16,17] where minimization approches using two constraints have been introduced.…”

Section: Introductionmentioning

confidence: 99%