Scattering threshold for the focusing nonlinear Klein–Gordon equation

Abstract

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig and Merle for the H 1 critical wave and Schrödinger equations. Our result includes the H 1 critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger-Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.

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

“…They also minimize the static energy 8) among all non-trivial static solutions. The work of Kenig, Merle [9,10] and Duyckaerts, Merle [5,6] allows for a characterization of the global-in-time behavior of solutions with E( u) ≤ J(W ).…”

Energy scattering for a Klein–Gordon equation with a cubic convolution

“…|u(t, x)| 2 |u(t, y)| 2 |x − y| γ dxdy ≡E(u 0 , u 1 ), and the momentum P (u)(t) = The scattering theory for the Klein-Gordon equation with f (u) = µ|u| p−1 u has been intensively studied in [4], [5], [9], [12], [34] and [36]. For µ = 1 and…”

“…Finally K. Nakanishi [34] obtained the scattering results for the critical case by the strategy of induction on energy [7] and a new Morawetztype estimate. And recently, S. Ibrahim, N. Masmoudi and K. Nakanishi [12,13] utilized the concentration compactness ideas to give the scattering threshold for the focusing nonlinear Klein-Gordon equation. Their method also works for the defocusing case.…”

“…Unfortunately, the type of virial identify in [37] can not work for the focusing case. Inspired by the method in [12] and [22], we get over this difficulty by some new variational framework and the profile decomposition under the restriction that the initial data is radial. Now we state our first result Theorem 1.1.…”

“…Moreover, this solution satisfies some strong compactness properties. This is completed in Section 6 where we utilize the profile decomposition that was established in [12], and a strategy introduced by Kenig and Merle [15]. We consider a virial-type identity in the direction orthogonal to the momentum vector following the technique [37] in the defocusing case to obtain a contradiction.…”