A note on the -critical inhomogeneous nonlinear Schrödinger equation

  • JinMyong An

    Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
  • JinMyong Kim

    Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
A note on the $H^{s}$-critical inhomogeneous nonlinear Schrödinger equation cover

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Abstract

In this paper, we consider the Cauchy problem for the -critical inhomogeneous nonlinear Schrödinger (INLS) equation

where , , and is a nonlinear function that behaves like with and . First, we establish the local well-posedness as well as the small data global well-posedness in for the -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 -critical INLS equation. Our results about the well-posedness and standard continuous dependence for the -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 and . Based on the local well-posedness in , we finally establish the blow-up criteria for -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.

Cite this article

JinMyong An, JinMyong Kim, A note on the -critical inhomogeneous nonlinear Schrödinger equation. Z. Anal. Anwend. 42 (2023), no. 3/4, pp. 403–433

DOI 10.4171/ZAA/1745