Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity

Abstract

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation i∂tu + Δu = −|x|b|u|p−1u in the radial Sobolev space Hr1(RN), where b > 0 and p > 1. We show the global existence and energy scattering in the intercritical regime, i.e., p>N+4+2bN and p<N+2+2bN−2 if N ≥ 3. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo–Nirenberg-type inequalities. Our proofs are based on improved Gagliardo–Nirenberg inequalities, the Morawetz–Sobolev approach of Dodson and Murphy [Proc. Am. Math. Soc. 145(11), 4859–4867 (2017)], radial Sobolev embeddings, and localized virial estimates.