Abstract. We consider the supercritical inhomogeneous nonlinear Schrödinger equation (INLS) i∂tuwhere (2 − b)/N < σ < (2 − b)/(N − 2) and 0 < b < min{2, N }. We prove a Gagliardo-Nirenberg type estimate and use it to establish sufficient conditions for global existence and blow-up in H 1 (R N ).
The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger equationthen the corresponding solution is global and scatters in H 1 (R 3 ). Our proof is based in the ideas introduced by Kenig-Merle [20] in their study of the energy-critical NLS and Holmer-Roudenko [17] for the radial 3D cubic NLS.
We study the local well-posedness of the initial-value problem for the nonlinear "good" Boussinesq equation with data in Sobolev spaces H s for negative indices of s.
We consider the generalized Korteweg-de Vries (gKdV) equation ∂tu + ∂ 3x u + µ∂x(u k+1 ) = 0, where k ≥ 5 is an integer number and µ = ±1. In the focusing case (µ = 1), we show that if the initial data u 0 belongs to, where M (u) and E(u) are the mass and energy, then the corresponding solution is global in H 1 (R). Here, s k = (k−4) 2kand Q is the ground state solution corresponding to the gKdV equation. In the defocusing case (µ = −1), if k is even, we prove that the Cauchy problem is globally well-posed in the Sobolev spaces H s (R), s > 4(k−1) 5k .
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