Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Global dynamics of the nonradial energy-critical wave equation above the ground state energy

Pages: 2423 - 2450, Volume 33, Issue 6, June 2013      doi:10.3934/dcds.2013.33.2423

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Joachim Krieger - Bâtiment des Mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland (email)
Kenji Nakanishi - Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (email)
Wilhelm Schlag - Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615, United States (email)

Abstract: In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions $3$ and $5$ assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in $\dot H^{1}\times L^{2}$ with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as $t → ±∞$.

Keywords:  Critical wave equation, blowup, scattering, stability, invariant manifold.
Mathematics Subject Classification:  35L05, 35B40.

Received: January 2012;      Revised: August 2012;      Available Online: December 2012.