Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

Abstract

<p style='text-indent:20px;'>In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS)</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta^{2} u = \lambda |x|^{-b}|u|^{\sigma}u, u(0) = u_{0} \in H^{s} (\mathbb R^{d}), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda \in \mathbb R $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d\in \mathbb N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<s<\min \{2+\frac{d}{2}, \frac{3}{2}d\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ 0<b<\min\{4, d, \frac{3}{2}d-s, \frac{d}{2}+2-s\} $\end{document}</tex-math></inline-formula>. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in <inline-formula><tex-math id="M5">\begin{document}$ H^{s}(\mathbb R^{d}) $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M6">\begin{document}$ \frac{8-2b}{d}<\sigma< \sigma_{c}(s) $\end{document}</tex-math></inline-formula> and the initial data is sufficiently small, where <inline-formula><tex-math id="M7">\begin{document}$ \sigma_{c}(s) = \frac{8-2b}{d-2s} $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M8">\begin{document}$ s<\frac{d}{2} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \sigma_{c}(s) = \infty $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M10">\begin{document}$ s\ge \frac{d}{2} $\end{document}</tex-math></inline-formula>.</p>

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

“…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)…”

“…Since 0 ≤ s c ≤ sc ≤ s, we can see that H s → Ḣsc ∩ Ḣ sc by using Lemmas 3.4 and 3.6. This implies that the smallness condition on the initial data u 0 in Theorem 1.3 is weaker than that of [7], where the global well-posedness of (1) was established in the case that ∥u 0 ∥ H s is sufficiently small. This paper is organized as follows.…”