On the Focusing Energy-Critical 3D Quintic Inhomogeneous NLS

On the focusing energy-critical inhomogeneous NLS: weighted space approach

“…We say that (1.1) is locally well-posed if there exists a maximal existence time interval I * such that there exists a unique solution u ∈ C(I * ; Ḣ1 ) and u depends continuously on the initial data. The local well-posedness (LWP) can be usually shown by a contraction argument based on the Strichartz estimates [4,8,6,15]. In this paper the L q0 t L r0 x (|x| −r0 γ * )-norm controls our whole contraction argument.…”

“…The aim of this paper is to establish a global theory for radial solutions: the global well-posedness (GWP), the scattering, and the finite time blowup to (1.1). In the previous papers [8,6] the authors considered a global theory for g with 0 ≤ b < 4 3 which was shown by a concrete concentration-compactness argument based on the local theory and profile decomposition. The restriction of index b is due to the lack of local theory of (1.1).…”

“…In order to handle the g with 4 3 ≤ b < 3 2 we develop an improved local theory, which consists of LWP and long-time perturbation, and develop a new profile decomposition based on the weighted space L q0 t L r0 x (|x| −r0 γ * ). The global theory can be shown straightforwardly by the concentration-compactness argument of [13,8,6].…”

“…

This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results [8,6] the authors considered the same problems with the spatial inhomogeneity coefficient g such that g(x) ∼ |x| −b for 0 ≤ b < 43 . Here we extend the inhomogeneous index b up to 3 2 .

…”

Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

“…Their results are given for N ≥ 3, 0 ≤ s < 1/3 or s = 1 and under some restrictive hypotheses on α and b. See also [9] for radially symmetric initial data inḢ 1 (R 3 ). It seems that the spaces used in [20,13,29,30] are not the naturel one to give the complete local well-posedness theory.…”