Blowup of [formula omitted] solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation

Abstract

We consider a class of focusing inhomogeneous nonlinear Schrödinger equation i∂tu+Δu+|x|−b|u|αu=0,u(0)=u0∈H1(Rd),with 0<b<min{2,d} and α⋆≤α<α⋆ where α⋆=4−2bd and α⋆=4−2bd−2 if d≥3 and α⋆=∞ if d=1,2. In the mass-critical case α=α⋆, we prove that if u0 has negative energy and satisfies either xu0∈L2 or u0 is radial with d≥2, then the corresponding solution blows up in finite time. Moreover, when d=1, we prove that if the initial data (not necessarily radial) has negative energy, then the corresponding solution blows up in finite time. In the mass and energy intercritical case α⋆<α<α⋆, we prove the finite time blowup for radial negative energy initial data as well as the finite time blowup below ground state for radial initial data in dimensions d≥2. This result extends the one of Farah in (J. Evol. Equ. 16: 193-208, 2016) where the blowup below ground state was proved for data in the virial space H1∩L2(|x|2dx) with d≥1.