Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity
Van Duong Dinh, Mohamed Majdoub, Tarek Saanouni
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.