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Journées équations aux dérivées partielles

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Kenji Nakanishi
Global dynamics beyond the ground state energy for nonlinear dispersive equations
Journées équations aux dérivées partielles (2012), Exp. No. 8, 6 p., doi: 10.5802/jedp.91
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Class. Math.: 35L70, 35Q55
Mots clés: nonlinear wave equations, nonlinear Schrödinger equation, nonlinear Klein-Gorond equation, solitons, scattering theory, blow-up, invariant manifolds

Résumé - Abstract

This is a brief introduction to the joint work with Wilhelm Schlag and Joachim Krieger on the global dynamics for nonlinear dispersive equations with unstable ground states. We prove that the center-stable and the center-unstable manifolds of the ground state solitons separate the energy space by the dynamical property into the scattering and the blow-up regions, respectively in positive time and in negative time. The transverse intersection of the two manifolds yields nine sets of global dynamics, which include stable transition from blow-up to scattering and vice versa.

Bibliographie

[1] K. Nakanishi J. Krieger & W. Schlag, “Global dynamics above the ground state energy for the one-dimensional NLKG equation”, to appear in Math. Z. , arXiv:1011.1776  MR 2968226 |  Zbl 1263.35002
[2] K. Nakanishi J. Krieger & W. Schlag, “Global dynamics away from the ground state for the energy-critical nonlinear wave equation”, to appear in Amer. J. Math. , arXiv:1010.3799  MR 3086065 |  Zbl pre06203653
[3] K. Nakanishi & W. Schlag, “Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation”, J. Differential Equations 250 (2011), p. 2299-2333  MR 2756065 |  Zbl 1213.35307
[4] K. Nakanishi & W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, European Mathematical Society, Zürich, 2011  MR 2847755 |  Zbl 1235.37002
[5] K. Nakanishi & W. Schlag, “Global dynamics above the ground state energy for the cubic NLS equation in 3D”, Calc. Var. Partial Differential Equations 44 (2012), p. 1-45  MR 2898769 |  Zbl 1237.35148
[6] K. Nakanishi & W. Schlag, “Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption”, Arch. Ration. Mech. Anal. 203 (2012), p. 809-851  MR 2928134 |  Zbl 1256.35138
[7] K. Nakanishi & W. Schlag, “Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation”, SIAM J. Math. Anal. 44 (2012), p. 1175-1210  MR 2914265 |  Zbl 1261.35037
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