On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

Abstract

In this paper, we consider the focusing mass-critical nonlinear fourth-order Schrödinger equation. We prove that blowup solutions to this equation with initial data in $H^{\gamma }({\mathbb{R}}^d),5\le d\le 7,\frac{56-3d+\sqrt{137d^2+1712d+3136}}{2(2d+32)}<\gamma <2$ 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 $u_0\in H^{\gamma }({\mathbb{R}}^d),5\le d\le 7,\frac{8d}{3d+8}<\gamma <2$ satisfying $\Vert u_0{\Vert }_{L^2({\mathbb{R}}^d)}<\Vert Q{\Vert }_{L^2({\mathbb{R}}^d)}$, where $Q$ is the solution to the ground state equation.

arXiv commentarXiv admin note: text overlap with arXiv:1706.06517