Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces
Volume 177 / 2024
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 $b$.
