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- Title
A note on the H<sup>s</sup>-critical inhomogeneous nonlinear Schrödinger equation.
- Authors
JinMyong An; JinMyong Kim
- Abstract
In this paper, we consider the Cauchy problem for the Hs-critical inhomogeneous nonlinear Schrödinger (INLS) equation ...; where n∈N, 0≤s< 2 n , 0<b<min{2,n−s,1+ 2 n−2s } and f(u) is a nonlinear function that behaves like λ∣u∣ σ u with λ∈C and σ= n−2s 4−2b . First, we establish the local well-posedness as well as the small data global well-posedness in H s (R n) for the H s -critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev–Lorentz spaces. Next, we obtain some standard continuous dependence results for the H s -critical INLS equation. Our results about the well-posedness and standard continuous dependence for the H s -critical INLS equation improve the ones of Aloui–Tayachi [Discrete Contin. Dyn. Syst. 41 (2021), 5409–5437] and An–Kim [Evol. Equ. Control Theory 12 (2023), 1039–1055] by extending the validity of s and b. Based on the local well-posedness in H 1 (R n), we finally establish the blow-up criteria for H 1-solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.
- Subjects
NONLINEAR Schrodinger equation; CAUCHY problem; BLOWING up (Algebraic geometry); NONLINEAR functions; SCHRODINGER equation
- Publication
Journal of Analysis & Its Applications / Zeitschrift für Analysis & ihre Anwendungen, 2023, Vol 42, Issue 3/4, p403
- ISSN
0232-2064
- Publication type
Article
- DOI
10.4171/ZAA/1745