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

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.