On blowup solutions to the focusing intercritical nonlinear fourth-order Schr\"odinger equation

TL;DR

This paper investigates the behavior of blowup solutions to a specific higher-order nonlinear Schrödinger equation, establishing profile decompositions, compactness, and concentration phenomena in critical Sobolev spaces.

Contribution

It introduces new analytical tools such as profile decomposition and variational characterization for the focusing intercritical fourth-order Schrödinger equation.

Findings

Profile decomposition in Sobolev spaces

Concentration of blowup solutions

Characterization of limiting blowup profiles

Abstract

In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schr\"odinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{\gamma_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{\gamma_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{\gamma_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{H}^{\gamma_{\text{c}}}$-norm.