Abstract
AbstractIn this paper we consider the mass-critical nonlinear Klein–Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle. The main novelty from the work of R. Killip, B. Stovall, and M. Visan (Trans. Amer. Math. Soc. 364, 2012) is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.
Funder
NSF of China
Australian Research Council
JSPS
Publisher
Springer Science and Business Media LLC
Reference41 articles.
1. Brenner, P.: On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations. Math. Z. 186, 383–391 (1984)
2. Cheng, X.: Scattering for the mass super-critical perturbations of the mass critical nolinear Schrödinger equations. Ill. J. Math. 64, 21–48 (2020)
3. Dodson, B.: Global well-posedness and scattering for the defocusing, $$L^2$$-critical nonlinear Schrödinger equation when $$d\ge 3$$. J. Amer. Math. Soc. 25, 429–463 (2012)
4. Dodson, B.: Global well-posedness and scattering for the defocusing, $$L^2$$-critical, nonlinear Schrödinger equation when $$d=2$$. Duke Math. J. 165, 3435–3516 (2016)
5. Dodson, B.: Global well-posedness and scattering for the defocusing, $$L^2$$-critical, nonlinear Schrödinger equation when $$d = 1$$. Amer. J. Math. 138, 531–569 (2016)