Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces
Year: 2024
Type: article
Abstract: In this paper, we study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] 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 $d\in \mathbb N$, $s>0$, $λ\in \mathbb R$ and $σ>0$. 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](2021) and Dinh [16](2018), where $f(u)$ is a nonlinear function that behaves like $λ|u|^σu$ with $λ\in \mathbb R$. These estimates are then applied to obtain the standard continuous dependence result for IBNLS equation with $0
Source: Colloquium Mathematicum
Authors: JinMyong An, YuIl Jo, JinMyong Kim
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4405541925