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

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.