Blow-up of solutions for semilinear fractional Schrödinger equations

Abstract

We consider the Cauchy problem in $\mathbb {R}^N$, $N \geq 1$, for the semi-linear Schr\"odinger equation with fractional Laplacian. We present the local well-posedness of solutions in $H^{{\alpha }/{2}}(\mathbb {R}^N)$, $0\lt \alpha \lt 2$. We prove a finite-time blow-up result, under suitable conditions on the initial data.