Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces
Abstract
We study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[ iu_{t} +\Delta ^{2} u=\lambda |x|^{-b}|u|^{\sigma }u,\quad u(0)=u_{0} \in H^{s} (\mathbb R^{d}), \] 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 $d\in \mathbb N$, $s \gt 0$, $\lambda \in \mathbb R$ and $\sigma \gt 0$. To arrive at this goal, we first obtain estimates of differences of values of a nonlinear function in fractional Sobolev spaces which generalize similar results of An–Kim (2022) and Dinh (2018), where the nonlinearity behaves like $\lambda |u|^{\sigma }u$ with $\lambda \in \mathbb R$. These estimates are then applied to obtain the standard continuous dependence result for the IBNLS equation with $0 \lt s \lt \min\{2+{d}/{2},{3d}/{2}\}$, $0 \lt b \lt \min\{4,d,{3d}/{2}-s,{d}/{2}+2-s\}$ and $0 \lt \sigma \lt \sigma_{c}(s)$, where $\sigma _{c}(s)=\frac{8-2b}{d-2s}$ if $s \lt {d}/{2}$, and $\sigma_{c}(s)=\infty $ if $s\ge {d}/{2}$. Our continuous dependence result generalizes that of Liu–Zhang (2021) by extending the range of $s$ and $b$.
