Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces
Tom 177 / 2024
Streszczenie
We study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation 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.
