Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

Abstract

We consider the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation in ${\mathbb{R}}^N$, $$i{\partial }_{t}u+{\Delta }^{2}u-|x{|}^{-b}|u{|}^{2\sigma }u=0,$$ where $\sigma >0$ and $b>0$. We first study the local well-posedness in ${\dot{H}}^{s_c}\cap {\dot{H}}^2$, for $N\ge 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in ${\dot{H}}^{s_c}\cap {\dot{H}}^2$ with $0\le s_c<2$. Finally, we study the phenomenon of $L^{{\sigma }_c}$-norm concentration for finite time blow up solutions with bounded ${\dot{H}}^{s_c}$-norm, where ${\sigma }_c=\frac{2N\sigma }{4-b}$. Our main tool is the compact embedding of ${\dot{L}}^p\cap {\dot{H}}^2$ into a weighted $L^{2\sigma +2}$ space, which may be seen of independent interest.

22 pages