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 Schrödinger (IBNLS) equation iu_t +\Delta^2 u=\lambda |x|^{-b}|u|^{\sigma}u,\quad u(0)=u_0 \in H^s (\mathbb{R}^d), where d\in \mathbb{N} , s\ge 0 , 0<b<4 , \sigma>0 and \lambda \in \mathbb{R} . Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in H^s(\mathbb{R}^d) if d\in \mathbb{N} , 0\le s <\min \{2+\nobreak\frac{d}{2},\frac{3}{2}d\} , 0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\} and 0<\sigma< \sigma_c(s) . Here \sigma_c(s)=\frac{8-2b}{d-2s} if s<\frac{d}{2} , and \sigma_c(s)=\infty if s\ge \frac{d}{2} . Our local well-posedness result improves the ones of Guzmán–Pastor (2020) and Liu–Zhang (2021) by extending the validity of s and b .

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

“…In Theorem 1.1, the result for the H s -critical case σ = σ c (s) with 0 ≤ s < d 2 is completely new. The result for the H s -subcritical case was already generalized in [6] to 0 ≤ s < min 2 + d 2 , 3 2 d and 0 < b < min 4, d, 3 2 d − s, d 2 + 2 − s . However, in this paper, we give the simple and unified proof in both of critical and 1 For s ∈ R, ⌈s⌉ denotes the minimal integer which is larger than or equal to s subcritical case.…”

“…The IBNLS equation (1) has attracted a lot of interest in recent years. See, for example, [6,7,10,11,22,23,29,35] and the references therein. Guzmán-Pastor [22] proved that (1) is locally well-posed in L 2 , if d ∈ N, 0 < b < min {4, d} and 0 < σ < σ c (0).…”

“…They also obtained the global well-posedness result in H 2 in the full range of mass-subcritical case and mass-critical cases 0 < σ ≤ 8−2b d . Very recently, the authors in [6,7] established the local and global well-posedness in H s for the IBNLS equation (1)…”