Global dynamics away from the ground state for the energy-critical nonlinear wave equation

Abstract

Abstract. We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground states and gets away from them in the energy space, we can classify its behavior into four cases, according to whether it blows up in finite time or scatters to zero, in forward or backward time direction. We prove that initial data for each case constitute a non-empty open set in the energy space.This is an extension of the recent results [15,16] by the latter two authors on the subcritical nonlinear Klein-Gordon and Schrödinger equations, except for the part of the center manifolds. The key step is to prove the "one-pass" theorem, which states that the transition from the scattering region to the blow-up region can take place at most once along each trajectory. The main new ingredients are the control of the scaling parameter and the blow-up characterization by Duyckaerts, Kenig and Merle [3,4].

A center-stable manifold for the energy-critical wave equation in ℝ3 in the symmetric setting

“…A number of results using alternate methods have been obtained for this equation by Kenig-Merle [13], Duyckaerts-Merle [10], Duyckaerts-Kenig-Merle [6][7][8], Krieger-Schlag-Tataru [18], Krieger-Nakanishi-Schlag [16,14], for solutions of energy less than that of the soliton φ or slightly above it. More recently, Duyckaerts-Kenig-Merle [9] classify all radial finite-energy solutions to (1.1).…”

Blow up and near soliton dynamics for the L 2 critical gKdV equation

“…Séminaire Laurent-Schwartz -EDP et applications Institut des hautes études scientifiques, 2011-2012 Exposé n o XXXVII, [1][2][3][4][5][6][7][8][9][10][11][12][13][14] XXXVII-1…”

“…Related rigidity theorems near the solitary wave were recently obtained by Nakanishi and Schlag [34], [35] and Krieger, Nakanishi and Schlag [13], for super critical wave and Schrödinger equations using the invariant set methods of Beresticky and Cazenave [1], the Kenig and Merle concentration compactness approach [9], the classification of minimal dynamics [4], [5], and a further "no return" lemma in the (Exit) regime. In the analogue of the (Exit) regime, this lemma shows that the solution cannot come back close to solitons and in fact scatters.…”

“…In the analogue of the (Exit) regime, this lemma shows that the solution cannot come back close to solitons and in fact scatters. In critical situations, such an analysis is more delicate and incomplete, see [13], and both the blow up statements and the no return lemma in [34], [35] rely on a specific algebraic structure -the virial identity -which does not exist for (gKdV).…”

“…(i) Blow up in finite time: for any ν > 11 13 , there exists u ∈ C((0, T 0 ), H 1 ) solution of (1) blowing up at t = 0 with…”