Scattering threshold for the focusing nonlinear Klein–Gordon equation

By S. Ibrahim, Nader Masmoudi, Kenji Nakanishi. 2010

162 citations recorded by the source corpus.

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of KenigMerle for the H critical wave and Schrödinger equations. Our result includes the H 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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7d992fae7816fbf9ddd5b6fc3e3a2c9d7f61077c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Global existence for the Euler-Maxwell system (2011, unknown): “We need the following Strichartz estimate for the Klein-Gordon equation (see for instance Ibrahim, Masmoudi and Nakanishi [20]): if ǫ > 0 and 0 ≤ δ ≤ 13 , ∥∥∥∥ ∫ t 0 eF (s) ds ∥∥∥∥ 2 .”
Adams' Inequality with the Exact Growth Condition in ℝ4 (2014, background): “The goal of this paper is to give the optimal growth rate of the exponential -type function when the problem is considered in the whole space R4.”
Trudinger-Moser Inequalities with the Exact Growth Condition in ℝ N and Applications (2015, methodology): “In order to overcome the possible failure of the Palais-Smale compactness condition, there is a common approach that relate [6, 18, 19, 28], and it involves a constrained minimization problem through the Pohozaev identity.”
Global well-posedness and scattering for nonlinear Schr\"odinger equations with combined nonlinearities in the radial case (2014, methodology): “They established the linear proﬁle decomposition in H 1 ( R d ) in the spirt of [13].”
On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces (2010, background): “See for example [3, 4, 16, 17, 27, 28, 34, 35, 36, 38, 41, 43, 44, 48, 49, 52] and the book [54] for further discussion and references.”
Equivalent Moser type inequalities in R2 and the zero mass case (2014, unknown): “1); see [3, 25] and more recently [26, 27, 30, 34].”