TITLE:
On Instability of Standing Waves for the Hartree Equation with a Magnetic Field
AUTHORS:
Rong Pang
KEYWORDS:
Hartree Equation, Magnetic Schr?dinger Equation, Standing Waves, Strong Instability, Blow-Up
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
24,
2026
ABSTRACT: In this paper, we consider the strong instability of standing waves for the Hartree equation with a constant magnetic field. This equation arises as a mean-field model describing the evolution of many-body Bosonic systems. We first establish the variational characterization of ground states. Then, we prove that if
?
2
?
λ
2
S
ω
(
?
λ
)|
λ=1
≤0
, the ground state standing wave
e
iωt
?(
x
)
is strongly unstable by blow-up, where
S
ω
is the action functional, and
?
λ
(
x
)=
λ
3/2
?(
λx
)
is the
L
2
-invariant scaling.