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

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 $H^{\gamma} (\mathbb{R}^d), 5 \leq d \leq 7, \frac{56 - 3d + \sqrt{137 d^2 + 1712 d + 3136}}{2(2d + 32)} \lt \gamma \lt 2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in [35] 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 $u_0 \in H^{\gamma} (\mathbb{R}^d), 5 \leq d \leq 7, \frac{8d}{3d+8} \lt \gamma \lt 2$ satisfying ${\lVert u_0 \rVert}_{L^2(\mathbb{R}^d)} \lt {\lVert Q \rVert}_{L^2(\mathbb{R}^d)}$, where $Q$ is the solution to the ground state equation.

Keywords

blowup, nonlinear fourth-order Schrödinger, global well-posedness, almost conservation law

2010 Mathematics Subject Classification

35B44, 35G20, 35G25

Full Text (PDF format)

Received 11 July 2017

Published 8 September 2017