Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schr\"odinger equation

Abstract

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schr\"odinger equation with spatial inhomogeneity coefficient $K(x)$ behaves like $\left|x\right|^{-b}$ for $0<b<\min \left\{\frac{N}{2},4\right\} $. We show the local well-posedness in the whole $H^s$-subcritical case, with $0<s\le2$. 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 Schr\"odinger equations in Besov spaces.