Limit of the blow-up solution for the inhomogeneous nonlinear Schrödinger equation

Abstract

We study the H1 blow-up profile for the inhomogeneous nonlinear Schrödinger equation i∂tu=−Δu−|x|k|u|2σu,(t,x)∈R×RN,where k∈(−1,2N−2) and N≥3. We develop a new version of Gagliardo–Nirenberg inequality for σ∈[2+kN,2+kN−2] and k∈(−1,2N−2), and show that for the L2-critical exponent σ=2+kN, u(t) has no L2-limit as t→T∗ when ‖u(t)‖H1 blows up at T∗. Moreover, we investigate L2 concentration at the origin in the radial case. Additionally, if 2+kN<σ<min{2,2+kN−2,2(1+k)+N2N}, we show that there exists a unique u∗∈L2(RN) such that Γ(−t)u(t)→Γ(−T∗)u∗inLr(RN) (r∈[2,2∗)) as t→T∗. Our results extend the work for k=0 by F. Merle in earlier time.