Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaces

Abstract

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr¨odinger (IBNLS) equation where d ∈ N , s ≥ 0, 0 < b < 4, σ > 0 and λ ∈ R . Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in H s ( R d ) if d ∈ N , 0 ≤ s < min { 2 + d 2 , 32 d } , 0 < b < min { 4 , d, 32 d − s, d 2 + 2 − s } and 0 < σ < σ c ( s ). Here σ c ( s ) = 8 − 2 b d − 2 s if s < d 2 , and σ c ( s ) = ∞ if s ≥ d 2 . Our local well-posedness result improves the ones of Guzm´an-Pastor [Nonlinear Anal. Real World Appl. 56 (2020) 103174] and Liu-Zhang [J. Differential Equations 296 (2021) 335-368] by extending the validity of s and b . Mathematics Subject Classification (2020) . Primary 35Q55; Secondary 35A01.