On blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schr\"odinger equation

TL;DR

This paper investigates the behavior of blowup solutions to a focusing nonlinear fractional Schrödinger equation, establishing concentration phenomena and limiting profiles for solutions with critical Sobolev norms.

Contribution

It introduces a compactness lemma via profile decomposition and characterizes the concentration and limiting profiles of blowup solutions in the critical Sobolev space.

Findings

Blowup solutions exhibit concentration in the critical Sobolev space.

The limiting profile of blowup solutions is characterized.

A new compactness lemma for the fractional Schrödinger equation is established.

Abstract

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schr\"odinger equation \[ i\partial_t u -(-\Delta)^s u = -|u|^\alpha u, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<\alpha<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. As a result, we obtain the $\dot{H}^{s_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{s_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{H}^{s_{\text{c}}}$-norm.