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

Abstract

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< γ < 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 R 4 . We also prove that the equation is globally well-posed with initial datawhere Q is the solution to the ground state equation.

Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations

“…Thanks to dispersive estimates (2.4) and the abstract theory of Keel-Tao [14], we have the following Strichartz estimates. Proposition 2.2 (Strichartz estimates [6,21]). Let µ ≥ 0 and I ⊂ R be an interval.…”

“…Dynamical properties such as mass-concentration and limiting profile of blow-up H 2 -solutions were studied by Zhu-Yang-Zhang [28] and the author [8]. Dynamical properties of blow-up solutions below the energy space were studied in [6,29].…”

Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space

“…The proof relies on the combination of the I-method and a new interaction Morawetz inequality. Recently, in [Din3] the author considered the defocusing cubic higher-order Schrödinger equation including the cubic fourth-order Schrödinger equation, and showed that the (NL4S) with ν = 3 is globally well-posed in H γ (R 4 ) with 60 53 < γ < 2. The argument makes use of the I-method and the bilinear Strichartz estimate.…”

On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation

“…The equation (1.1) is a special case of (1.3) with ǫ = 0 and µ = −1. The study of nonlinear fourth-order Schrödinger equations (1.3) has attracted a lot of interest in the past several years (see [26], [27], [13], [14], [16], [22], [23], [24], [8], [9] and references therein).…”

“…without radially symmetric assumption). For dynamical properties of blowup solutions with low regularity initial data, we refer the reader to [32] and [9].…”