Blow-up of radial solutions for the intercritical inhomogeneous NLS equation

Abstract

We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in RNi∂tu+Δu+|x|−b|u|2σu=0, where N≥3, 0<b<min⁡{N2,2} and 2−bN<σ<2−bN−2. The scaling invariant Sobolev space is H˙sc with sc=N2−2−b2σ. The restriction on σ implies 0<sc<1 and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let u0∈H˙sc∩H˙1 be a radial initial data and u(t) the corresponding solution to the INLS equation. We first show that if E[u0]≤0, then the maximal time of existence of the solution u(t) is finite. Also, for all radially symmetric solution of the INLS equation with finite maximal time of existence T⁎>0, then limsupt→T⁎‖u(t)‖H˙sc=+∞. Moreover, under an additional assumption and recalling that H˙sc⊂Lσc with σc=2Nσ2−b, we can in fact deduce, for some γ=γ(N,σ,b)>0, the following lower bound for the blow-up ratec‖u(t)‖H˙sc≥‖u(t)‖Lσc≥|log⁡(T−t)|γ,ast→T⁎. The proof is based on the ideas introduced for the L2 super critical nonlinear Schrödinger equation in the work of Merle and Raphaël [14] and here we extend their results to the INLS setting.