On blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation

Abstract

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d\ge 2,\frac{d}{2d-1}\le s<1$, $\frac{4s}{d}<\alpha <\frac{4s}{d-2s}$ and $u_0\in {\dot{H}}^{s_{\text{c}}}\cap {\dot{H}}^s$ 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.

arXiv comment27 pages, revised version, to appear Journal Mathematical Physics. arXiv admin note: text overlap with arXiv:1801.08866, arXiv:1801.08049