Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation

Abstract

In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation i∂tu+Δu+|x|−b|u|pu=0,(t,x)∈R×RNwith 0<b<min{2,N} and 4−2bN<p<4−2bN−2. Assume that u0∈H1(RN) and beyond the ground state threshold, then we prove the following two statements, (1) when 4−2bN<p<min{4N,4−2bN−2}, or p=4N when b∈(0,4N), then the corresponding solution blows up in finite time; (2) when 4N<p<4−2bN−2, we prove the finite or infinite time blow-up. Moreover, we can further obtain a precise lower bound of infinite time blow-up rate, that is supt∈[0,T]‖∇u(t)‖L2≳Tκ,for someκ>0.To our knowledge, the statement (1) establishes the first finite time blow-up result for this equation in the intercritical case when the initial data u0 does not have finite variance and is non-radial. The statement (2) gives the first result for the infinite time blow-up rate for this equation.