Global dynamics beyond the ground state energy for nonlinear dispersive equations
Kenji Nakanishi1
1 Department of Mathematics Kyoto University Kyoto 606-8502 Japan
Journées équations aux dérivées partielles (2012), article no. 8, 6 p.

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.

DOI : 10.5802/jedp.91
Classification : 35L70, 35Q55
Keywords: nonlinear wave equations, nonlinear Schrödinger equation, nonlinear Klein-Gorond equation, solitons, scattering theory, blow-up, invariant manifolds
Kenji Nakanishi 1

1 Department of Mathematics Kyoto University Kyoto 606-8502 Japan 
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Kenji Nakanishi. Global dynamics beyond the ground state energy for nonlinear dispersive equations. Journées équations aux dérivées partielles (2012), article  no. 8, 6 p. doi : 10.5802/jedp.91. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.91/

[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 | Zbl

[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

[3] K. Nakanishi; W. Schlag Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations, Volume 250 (2011), pp. 2299-2333 | MR | Zbl

[4] K. Nakanishi; W. Schlag Invariant manifolds and dispersive Hamiltonian evolution equations, European Mathematical Society, Zürich, 2011 | MR | Zbl

[5] K. Nakanishi; W. Schlag Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, Volume 44 (2012), pp. 1-45 | MR | Zbl

[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., Volume 203 (2012), pp. 809-851 | MR | Zbl

[7] K. Nakanishi; W. Schlag Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation, SIAM J. Math. Anal., Volume 44 (2012), pp. 1175-1210 | MR | Zbl

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