We consider the focusing energy-critical inhomogeneous nonlinear Schrödinger equation:3 , and p = 5 − 2b. On the road map of Kenig-Merle [22] we show the global well-posedness and scattering of radial solutions under energy condition, and rigidity condition −bg(x) ≤ x · ∇g(x). We also provide sharp finite time blowup results for non-radial and radial solutions. For this we utilize the localized virial identity.2010 Mathematics Subject Classification. M35Q55, 35Q40.
We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results [7,6]; we establish global well-posedness and scattering for large data with a certain condition. Firstly we investigate the long-time behavior of solutions to the Dirac equation satisfies good control provided that a particular dispersive norm of solutions is bounded. The key of our proof relies on modifying multilinear estimates obtained in our previous papers.Secondly, we obtain large data scattering by exploiting the Majorana condition.
In this paper, we consider the Cauchy problem of regularity and uniqueness of the Chern-Simons-Dirac system in the Coulomb gauge for initial data in B 0 2,1 . The novelty of this paper is on proving almost critical regularity by using the fully localization of space-time Fourier side and bilinear estimates given by Selberg [17]. We also prove the Dirac spinor flow of Chern-Simons-Dirac system cannot be C 3 at the origin in H s if s < 0. 1 2 < ∞}. Our main result is stated as follows.Theorem 1.1. Suppose that ψ 0 ∈ B 0 2,1 . Then there exists T = T ( ψ 0 B 0 2,1 , m) > 0 such that there exists unique solution ψ ∈ C((−T, T ); B 0 2,1 ) of (1.1), which depends continuously on the initial data.
Abstract. We consider the mass concentration phenomenon for the L 2 -critical nonlinear Schrödinger equations of higher orders. We show that any solution u to, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as α increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.
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