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Register a new userScattering for the quadratic Klein-Gordon equations
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We study the scattering problems for the quadratic Klein-Gordon equations with radial initial data in the energy space. For 3D, we prove small data scattering, and for 4D, we prove large data scattering with mass below the ground state.
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We prove global existence backwards from the scattering data posed at infinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying the weak null condition. The asymptotics of the solutions to the Maxwell Klein Gordon equations in Lorenz gauge were shown to be wave like at null infinity and homogeneous towards timelike infinity in arXiv:1803.11086 and expressed in terms of radiation fields, and thus our scattering data will be given in the form of radiation fields in the backward problem. We give a refinement of the asymptotics results in arXiv:1803.11086, and then making use of this refinement, we find a global solution which attains the prescribed scattering data at infinity. Our work starts from the approach in [22] and is more delicate since it involves nonlinearities with fewer derivatives. Our result corresponds to existence of scattering states in the scattering theory. The method of proof relies on a suitable construction of the approximate solution from the scattering data, a weighted conformal Morawetz energy estimate and a spacetime version of Hardy inequality.
In this paper we consider the real-valued 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 as R. Killip, B. Stovall, and M. Visan [Trans. Amer. Math. Soc. 364 (2012)]. The main new novelty is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrodinger equation when the nonlinearity is not algebraic.
This article resolves some errors in the paper Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis & PDE 4 (2011) no. 3, 405-460. The errors are in the energy-critical cases in two and higher dimensions.
On the three dimensional Euclidean space, for data with finite energy, it is well-known that the Maxwell-Klein-Gordon equations admit global solutions. However, the asymptotic behaviours of the solutions for the data with non-vanishing charge and arbitrary large size are unknown. It is conjectured that the solutions disperse as linear waves and enjoy the so-called peeling properties for pointwise estimates. We provide a gauge independent proof of the conjecture.
We revisit the scattering problems for the 2D mass super-critical Schr{o}dinger and Klein-Gordon equations with radial data below the ground state in the energy space. We give an alternative proof of energy scattering for both defocusing and focusing cases using the ideas of Dodson-Murphy citep{dodson2017new-radial}. Our results also include the exponential type nonlinearities which seems to be new for the focusing exponential NLS.
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