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

Abstract

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in $${\mathbb {R}}^N$$ , $$\begin{aligned} i \partial _t u +\Delta ^2 u -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$ 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.