Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation
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- by Slim Ibrahim, Nader Masmoudi and Kenji Nakanishi PDF
- Trans. Amer. Math. Soc. 366 (2014), 5653-5669 Request permission
Abstract:
We study global dynamics for the focusing nonlinear Klein-Gordon equation with the energy-critical nonlinearity in two or higher dimensions when the energy equals the threshold given by the ground state of a mass-shifted equation, and prove that the solutions are divided into scattering and blowup. In short, the Kenig-Merle scattering/blowup dichotomy extends to the threshold energy in the case of mass-shift for the critical nonlinear Klein-Gordon equation.References
- Thomas Duyckaerts, Carlos Kenig, and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1389–1454. MR 2966655, DOI 10.4171/JEMS/336
- Thomas Duyckaerts and Frank Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), no. 6, 1787–1840. MR 2491692, DOI 10.1007/s00039-009-0707-x
- Thomas Duyckaerts and Frank Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP , posted on (2008), Art ID rpn002, 67. MR 2470571, DOI 10.1093/imrp/rpn002
- J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys. 123 (1989), no. 4, 535–573. MR 1006294
- Slim Ibrahim, Mohamed Majdoub, and Nader Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math. 59 (2006), no. 11, 1639–1658. MR 2254447, DOI 10.1002/cpa.20127
- Slim Ibrahim, Mohamed Majdoub, and Nader Masmoudi, Well- and ill-posedness issues for energy supercritical waves, Anal. PDE 4 (2011), no. 2, 341–367. MR 2859857, DOI 10.2140/apde.2011.4.341
- Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi, and Kenji Nakanishi, Scattering for the two-dimensional energy-critical wave equation, Duke Math. J. 150 (2009), no. 2, 287–329. MR 2569615, DOI 10.1215/00127094-2009-053
- Slim Ibrahim, Nader Masmoudi, and Kenji Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE 4 (2011), no. 3, 405–460. MR 2872122, DOI 10.2140/apde.2011.4.405
- S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger-Moser inequality on the whole plane with the exact growth condition, submitted.
- Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675. MR 2257393, DOI 10.1007/s00222-006-0011-4
- Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), no. 2, 147–212. MR 2461508, DOI 10.1007/s11511-008-0031-6
- L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273–303. MR 402291, DOI 10.1007/BF02761595
Additional Information
- Slim Ibrahim
- Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3060 STN CSC, Victoria, British Columbia, Canada V8P 5C3
- MR Author ID: 646053
- Email: ibrahim@math.uvic.ca
- Nader Masmoudi
- Affiliation: The Courant Institute for Mathematical Sciences, New York University, New York, New York 10012
- MR Author ID: 620387
- Email: masmoudi@courant.nyu.edu
- Kenji Nakanishi
- Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 643074
- Email: n-kenji@math.kyoto-u.ac.jp
- Received by editor(s): October 7, 2011
- Received by editor(s) in revised form: April 10, 2012
- Published electronically: May 20, 2014
- Additional Notes: The first author was partially supported by NSERC# 371637-2009 grant and a start up fund from the University of Victoria
The second author was partially supported by an NSF Grant DMS-0703145 - © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 5653-5669
- MSC (2010): Primary 35L70, 35B40, 35B44, 47J30
- DOI: https://doi.org/10.1090/S0002-9947-2014-05852-2
- MathSciNet review: 3256178