Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation. (July 2023) - Bai, Ruobing; Li, Bing
Abstract: In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation i ∂ t u + Δ u + | x | − b | u | p u = 0, ( t, x ) ∈ R × R N with 0 < b < min 2, N and 4 − 2 b N < p < 4 − 2 b N − 2 . Assume that u 0 ∈ H 1 ( R N ) and beyond the ground state threshold, then we prove the following two statements, (1) when 4 − 2 b N < p < min 4 N, 4 − 2 b N − 2 , or p = 4 N when b ∈ ( 0, 4 N ), then the corresponding solution blows up in finite time; (2) when 4 N < p < 4 − 2 b N − 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 sup t ∈ [ 0, T ] ‖ ∇ u ( t ) ‖ L 2 ≳ T κ, for some κ > 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 u 0 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.