On the focusing mass-critical nonlinear fourth-order Schr\"odinger equation below the energy space

Abstract

In this paper, we consider the focusing mass-critical nonlinear fourth-order Schr\"odinger equation. We prove that blowup solutions to this equation with initial data in $H^\gamma(\mathbb{R}^d), 5\leq d \leq 7, \frac{56-3d+\sqrt{137d^2+1712d+3136}}{2(2d+32)} <\gamma<2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in \cite{ZhuYangZhang11} where Zhu-Yang-Zhang studied the formation of singularity for the equation with rough initial data in $\mathbb{R}^4$. We also prove that the equation is globally well-posed with initial data $u_0 \in H^\gamma(\mathbb{R}^d), 5\leq d \leq 7, \frac{8d}{3d+8}<\gamma<2$ satisfying $\|u_0\|_{L^2(\mathbb{R}^d)} <\|Q\|_{L^2(\mathbb{R}^d)}$, where $Q$ is the solution to the ground state equation.