In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient $K(x)$ behaves like ${\left|x\right|}^{-b}$ for $0<b<\min \{\frac{N}{2},4\}$. We show the local well-posedness in the whole $H^s$-subcritical case, with $0<s\le 2$. The difficulties of this problem come from the singularity of $K(x)$ and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces.

中文翻译：

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Besov 空间中的双线性 Strichartz 类型估计及其对非齐次非线性双调和 Schrödinger 方程的应用

在本文中，我们考虑了具有空间不均匀系数的非齐次非线性双调和薛定谔方程的适定性 $钾(X)$ 表现得像 ${\left|X\right|}^{-乙}$ 为了 $0<乙<\mathrm{分钟}\{\frac{N}{2},4\}$. 我们在整体上表现出局部的适中性$H^{秒}$- 亚临界情况，有 $0<秒\le 2$. 这个问题的难点来自于$钾(X)$以及非线性项缺乏可微性。为了解决这个问题，我们导出了 Besov 空间中非线性双调和薛定谔方程的双线性 Strichartz 类型估计。