Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation
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概要
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The analysis of global dynamics of nonlinear dispersive equationshas a long history starting from small solutions. In this paper we study thefocusing, cubic, nonlinear Klein-Gordon equation in R3 with large radial data inthe energy space. This equation admits a unique positive stationary solution Q,called the ground state. In 1975 Payne and Sattinger showed that solutions u(t)with energy E[u, u˙ ] strictly below that of the ground state are divided into twoclasses, depending on a suitable functional K(u): If K(u) < 0, then one has finitetime blow-up, if K(u) ≥ 0 global existence; moreover, these sets are invariantunder the flow. Recently, Ibrahim, Masmoudi and the first author [22] improvedthis result by establishing scattering to zero for K[u] ≥ 0 by means of a variantof the Kenig-Merle method [25], [26]. In this paper we go slightly beyond theground state energy and we give a complete description of the evolution in thatcase. For example, in a small neighborhood of Q one encounters the followingtrichotomy: on one side of a center-stable manifold one has finite-time blow-upfor t ≥ 0, on the other side scattering to zero, and on the manifold itself one hasscattering to Q, both as t → +∞. In total, the class of data with energy at mostslightly above that of Q is divided into nine disjoint nonempty sets each displayingdifferent asymptotic behavior as t → ±∞, which includes solutions blowing up inone time direction and scattering to zero on the other. The analogue of thesolutions found by Duyckaerts, Merle [13], [14] for the energy critical wave andSchr¨odinger equations appear here as the unique one-dimensional stable/unstablemanifolds approaching ±Q exponentially as t → ∞ or t → −∞, respectively. Themain technical ingredient in our proof is a “one-pass” theorem which excludes theexistence of (almost) homoclinic orbits between Q (as well as −Q) and (almost)heteroclinic orbits connecting Q with −Q. In a companion paper [31] we establishanalogous properties for the NLS equation.
- 2011-03-01
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