On the global well-posedness of focusing energy-critical inhomogeneous NLS

Abstract

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.

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 .

…”

Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations

“…The energy-critical case s = 1 was also handled by the authors [19] for 0 < α < min{n/2, 2} and n ≥ 3. By this weighted norm approach, some related results in [7] for the focusing energy-critical case could be also improved in [9]. The gap 1/3 ≤ s < 1 was recently filled in [1] by utilizing the known Strichartz estimates [16] in Lorentz spaces L p,2 .…”