Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces (2305.17900v1)

Abstract: In this paper, we study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation [iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),] in the standard sense in $H^s$, i.e. in the sense that the local solution flow is continuous $H^s\to H^s$. Here $d\in \mathbb N$, $s>0$, $\lambda\in \mathbb R$ and $\sigma>0$. To arrive at this goal, we first obtain the estimates of the term $f(u)-f(v)$ in the fractional Sobolev spaces which generalize the similar results of An-Kim 5 and Dinh 16, where $f(u)$ is a nonlinear function that behaves like $\lambda |u|^{\sigma}u$ with $\lambda\in \mathbb R$. These estimates are then applied to obtain the standard continuous dependence result for IBNLS equation with $0<s <\min {2+\frac{d}{2},\frac{3}{2}d}$, $0<b<\min{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s}$ and $0<\sigma< \sigma_{c}(s)$, where $\sigma_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $\sigma_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. Our continuous dependence result generalizes that of Liu-Zhang 27 by extending the validity of $s$ and $b$.