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

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schr"odinger equation $$i{\mathrm{\partial }}_{t}u-(-\mathrm{\Delta }{)}^{s}u=-\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,u(0)=u_0,\text{on }[0,\mathrm{\infty })\times {\mathbb{R}}^d,$$ where $d\ge 2,\frac{d}{2d-1}\le s<1$, $\frac{4s}{d}<\alpha <\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{$\text{c}$}} \cap \dot{H}^s$isradialwiththecriticalSobolevexponent$s_{$\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}$}} \cap \dot{H}^s$\mathrm{.}Asaresult,weobtainthe$\dot{H}^{s_{$\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.