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.
Mots-clés : nonlinear wave equations, nonlinear Schrödinger equation, nonlinear Klein-Gorond equation, solitons, scattering theory, blow-up, invariant manifolds
@incollection{JEDP_2012____A8_0,
author = {Nakanishi, Kenji},
title = {Global dynamics beyond the ground state energy for nonlinear dispersive equations},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {8},
pages = {1--6},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2012},
doi = {10.5802/jedp.91},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.91/}
}
TY - JOUR AU - Nakanishi, Kenji TI - Global dynamics beyond the ground state energy for nonlinear dispersive equations JO - Journées équations aux dérivées partielles PY - 2012 SP - 1 EP - 6 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.91/ DO - 10.5802/jedp.91 LA - en ID - JEDP_2012____A8_0 ER -
%0 Journal Article %A Nakanishi, Kenji %T Global dynamics beyond the ground state energy for nonlinear dispersive equations %J Journées équations aux dérivées partielles %D 2012 %P 1-6 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.91/ %R 10.5802/jedp.91 %G en %F JEDP_2012____A8_0
Nakanishi, Kenji. 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. http://www.numdam.org/articles/10.5802/jedp.91/
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[2] 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] 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
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[5] 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] 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] 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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