Global Existence and Finite Time Blow-Up for the Nonlinear Schrödinger Hartree Equation with a Constant Magnetic Field

The paper addresses the global existence and finite-time blow-up phenomena for the nonlinear Schrödinger Hartree equation in the presence of a constant magnetic field, which is given by:

i

∂

t

ψ+

(

∇+iA

)

2

ψ−(

| x |

−1

∗

| ψ |

2

)ψ+

| ψ |

p

ψ=0

,

(

t,x

)∈

ℝ

+

×

ℝ

3

. This equation is considered in three-dimensional space. Under mass-critical and supercritical conditions, we determine the precise thresholds for global existence and finite-time blow-up in the

L

2

-supercritical regime, specifically for the range

4

3

<p<4

.

• Nonlinear Schrödinger Hartree Equation
• Global Existence
• Finite-Time Blow-Up

1. Introduction

Recently, we have studied a class of nonlinear Schrödinger Hartree equations featuring a constant magnetic field

(1.1) { i ∂ t ψ + ( ∇ + i A ) 2 ψ − ( | x | − 1 ∗ | ψ | 2 ) ψ + | ψ | p ψ = 0 ,       ( t , x ) ∈ ℝ + × ℝ 3 , ψ | t = 0 = ψ 0 .

where

(1.2) A ( x ) = b 2 ( − x 2 , x 1 , 0 ) ,       x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 .

A vector-valued potential is employed to model the influence of an external magnetic field

B = c u r l ( A ) = ( 0 , 0 , b ) ,     b ≠ 0.

The nonlinear Schrödinger equation (NLS) and its generalizations are widely used in quantum mechanics (see e.g., [1]), Bose-Einstein condensation (see e.g., [2][3]), and nonlinear optics. These equations combine three key mechanisms: the magnetic Laplace operator for quantum dynamics in external magnetic fields, the Hartree-type nonlocal term for long-range interactions, and a local power nonlinearity for short-range interactions.

For blow-up theory, Glassey [4] introduced the virial identity to show finite-time blow-up for solutions with negative initial energy. Weinstein [5] provided sharp interpolation estimates that define blow-up thresholds. These results have been extended to nonlocal nonlinearity by Feng [6], and others.

In global existence theory, Cazenave [7], and Sulem [8] developed well-posedness frameworks and a priori bounds. Lions [9] concentration-compactness principle helped prove the existence of ground states and analyze their dynamics. For equations with mixed nonlinearities, Feng, Zhang, and Zhu [10]-[12] studied how competing effects influence blow-up. Additionally, magnetic Schrödinger operators have been studied by Arioli [13], Esteban [14], and Kurata [15]; spectral properties under magnetic fields were explored by Helffer [16], and Bouclet [17] established Strichartz estimates in this setting.

Before presenting our findings in this regard, it is essential to revisit the local theory pertaining to Equation (1.1). The local well-posedness of Equation (1.1) for initial data within the space H A 1 ( ℝ 3 ) was proven by Cazenave and Esteban in [7][18]. Here the space H A 1 ( ℝ 3 ) is defined as

H A 1 ( ℝ 3 ) : = { f ∈ L 2 ( ℝ 3 ) : | ( ∇ + i A ) f | ∈ L 2 ( ℝ 3 ) }

is a Hilbert space endowed with a specific norm

‖ f ‖ H A 1 2 = ‖ ( ∇ + i A ) f ‖ L 2 2 + ‖ f ‖ L 2 2 .

More formally, when the parameter p is confined to the interval 0 < p < 4 and the initial data ψ 0 is selected from the Hilbert space H A 1 ( ℝ 3 ) , there exists a time T * ∈ ( 0 , ∞ ) , together with a uniquely defined maximal solution.

ψ ∈ C ( [ 0 , T * ) , H A 1 ( ℝ 3 ) ) ∩ C 1 ( [ 0 , T * ) , H A − 1 ( ℝ 3 ) ) ,

where H A − 1 ( ℝ 3 ) is the dual space of H A 1 ( ℝ 3 ) . For each ψ 0 ∈ H A 1 ( ℝ 3 ) , the solution to (1.1) exists on a maximal time interval ( − T * , T * ) . The maximal time of existence satisfies the blow-up alternative: if T * < ∞ (resp. T * < ∞ ), then

lim t ↗ T * ‖ ∇ u ( t ) ‖ L 2 = ∞     ( resp .     lim t ↘ − T * ‖ ∇ u ( t ) ‖ L 2 = ∞ ) .

Moreover there are conservation laws of mass and energy:

M ( ψ ( t ) ) = ‖ ψ ( t ) ‖ L 2 2 = M ( ψ 0 ) , E ( ψ ( t ) ) = 1 2 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 2 + 1 4 ∫ ℝ 3 ( 1 | x | ∗ | ψ ( t ) | 2 ) | ψ ( t ) | 2 d x − 1 p + 2 ‖ ψ ( t ) ‖ L p + 2 p + 2 = E ( ψ 0 )

for all t ∈ [ 0 , T * ) .

For stating the blow-up results associated with (1.1), let us define the subsequent Hilbert space

Σ A ( ℝ 3 ) : = { f ∈ H A 1 ( ℝ 3 ) : | x | f ∈ L 2 ( ℝ 3 ) } ,

equipped with the norm

‖ f ‖ Σ A 2 = ‖ ( ∇ + i A ) f ‖ L 2 2 + ‖ x f ‖ L 2 2 + ‖ f ‖ L 2 2 .

It is well-established that the space Σ A ( ℝ 3 ) coincides with the standard weighted Sobolev space

Σ ( ℝ 3 ) : = { f ∈ H 1 ( ℝ 3 ) : | x | f ∈ L 2 ( ℝ 3 ) } .

Owing to this property, we obtain the following useful identity:

(1.3) ‖ ( ∇ + i A ) f ‖ L 2 2 = ‖ ∇ f ‖ L 2 2 + b R ( f ) + b 2 4 ‖ ρ f ‖ L 2 2 , ρ : = x 1 2 + x 2 2

The structure of this paper is outlined below: Chapter 2 introduces the mathematical framework and lays out the relevant preliminary findings. Chapter 3 establishes precise thresholds for global existence and finite-time blow-up in the mass-critical and mass-supercritical regimes.

Theorem 1.1 [19][20] The set

4 3 < p < 4 , ψ 0 ∈ Σ A ( ℝ 3 ) and ψ : [ 0 , T * ) × ℝ 3 → ℂ be the correspondingsolution to (1.1). Then the solution blows up in finite time, i.e. T * < ∞ proved that one of the following conditions holds:

[20] E ( ψ 0 ) < 0 ;[19] E 0 ( ψ 0 ) < 0 ;[19] E 0 ( ψ 0 ) = 0 and Im ∫ x ⋅ ∇ ψ 0 ( x ) ψ ¯ 0 d x < 0 ;[19] E 0 ( ψ 0 ) > 0 and Im ∫ x ⋅ ∇ ψ 0 ( x ) ψ ¯ 0 d x < − 2 E 0 ( ψ 0 ) ‖ x ψ 0 ‖ L 2 .

Where

(1.4) E 0 ( f ) : = 1 2 ‖ ∇ f ‖ L 2 2 + b 2 8 ‖ ρ f ‖ L 2 2 + 1 4 ∫ ℝ 3 ( 1 | x | ∗ | f | 2 ) | f | 2 d x − 1 p + 2 ‖ f ‖ L p + 2 p + 2 .

Remark 1.1 Thanks to the conservation of angular momentum, i.e., R ( ψ ( t ) ) = R ( ψ 0 ) , we observe that

(1.5) E 0 ( ψ ( t ) ) = E 0 ( ψ 0 ) ,     ∀ t ∈ [ 0 , T * ) .

Theorem 1.2 Let p = 4 3 .

1) If ψ 0 ∈ H A 1 ( ℝ 3 ) satisfies ‖ ψ 0 ‖ L 2 < ‖ Q ‖ L 2 , where Q is the unique positive radial solution to

(1.6) − Δ Q + Q + 1 | x | ∗ | Q | 2 Q − | Q | p Q = 0.

then the corresponding solution to (1.1) exists globally in time, i.e. T * = ∞ .

2) For c > ‖ Q ‖ L 2 , there exists ψ 0 ∈ Σ A ( ℝ 3 ) such that the corresponding solution to (1.1) with initial data ψ | t = 0 = ψ 0 blow up in finite time, i.e. T * = ∞ .

Lemma1.2 The equation

− Δ Q + Q + 1 | x | ∗ | Q | 2 Q − | Q | p Q = 0 ,     x ∈ ℝ 3

admits a unique positive radial solution Q > 0 in H 1 ( ℝ 3 ) satisfying Q ( x ) → 0 as | x | → ∞ , and Q ∈ Σ A ( ℝ 3 ) .

proof. The right-hand side is a spherically symmetric nonlinear term that is locally Lipschitz continuous with respect to Q , satisfying the core conditions of the integral moving plane method. By the radial symmetry conclusion and the rotational invariance of the equation, all positive solutions are radially symmetric about the origin, i.e., Q = Q ( r ) ( r = | x | ) , and the radial solution is strictly monotonic with Q ′ ( r ) < 0 for all r > 0 . Thus, it suffices to prove the uniqueness in the radial Sobolev space H A 1 ( ℝ 3 ) . For the radial function Q = Q ( r ) , the radialform of the three-dimensional Laplace operator is Δ Q = Q ″ + 2 r Q ′ . Denote the radial nonlocal term by

V ( r ) = ( 1 | x | ∗ | Q | 2 ) ( r ) = ∫ 0 ∞ s 2 Q ( s ) 2 max ( r , s ) d s .

Then the original equation is reduced to a second-order nonlinear ordinary differential equation (ODE):

Q ″ + 2 r Q ′ − Q + V ( r ) Q − Q 7 3 = 0 ,     r > 0 ,

endowed with the regularity and decay conditions: Q ′ ( 0 ) = 0 , Q ( r ) > 0 for all r > 0 , and Q ( r ) → 0 as r → ∞ .

By the local existence and uniqueness theorem for ODEs, there exists a unique local solution Q ( ⋅ ; a ) for any initial value Q ( 0 ) = a > 0 . Combining with the Pohozaev identity in [21], this local solution can be extended to all r > 0 , and Q ( r ) decays exponentially as Q ( r ) ~ C e − r r for r → ∞ , hence Q ∈ Σ A ( ℝ 3 ) .Define the radial energy density matching the form of the energy functional in this paper:

E ( r ) = 1 2 ( Q ′ ) 2 + 1 2 Q 2 + 1 4 V ( r ) Q 2 − 3 10 Q 10 3 .

Taking the derivative and substituting the transformed radial ODE: Q ″ + Q + V ( r ) Q − Q 7 3 = − 2 r Q ′ , we simplify to obtain

E ′ ( r ) = Q ′ ( − 2 r Q ′ ) + 1 2 V ′ ( r ) Q 2 = − 2 r ( Q ′ ) 2 + 1 2 V ′ ( r ) Q 2 .

From Q ′ ( r ) ≤ 0 and the monotonicity of the nonlocal term V ′ ( r ) = − r 2 Q ( r ) 2 ≤ 0 , we directly get E ′ ( r ) ≤ 0 for all r > 0 , which means the energy density is non-increasing along the radial direction. Suppose there exist two distinct positive radial decaying solutions Q 1 = Q ( ⋅ ; a 1 ) and Q 2 = Q ( ⋅ ; a 2 ) satisfying 0 < a 1 < a 2 and the above regularity and decay conditions. At r = 0 , Q ′ 1 ( 0 ) = Q ′ 2 ( 0 ) = 0 , and the energy density is given by E i ( 0 ) = 1 2 a i 2 + 1 4 V i ( 0 ) a i 2 − 3 10 a i 10 3 . Define φ ( t ) = 1 2 t 2 + 1 4 V ( 0 ) t 2 − 3 10 t 10 3 , which is strictly increasing for t > 0 , thus E 1 ( 0 ) < E 2 ( 0 ) . Since both Q 1 and Q 2 decay exponentially to 0, we have lim r → ∞ E 1 ( r ) = lim r → ∞ E 2 ( r ) = 0 . Combining the monotonicity of the energy density, we obtain E 1 ( r ) ≤ E 1 ( 0 ) < E 2 ( 0 ) ≤ E 2 ( r ) for all r > 0 . Let w ( r ) = Q 2 ( r ) − Q 1 ( r ) , then w ( 0 ) = a 2 − a 1 > 0 and w ′ ( 0 ) = 0 . Substitute Q 1 and Q 2 into the radial ODE and subtract the two equations; by the mean value theorem, we have

w ″ + 2 r w ′ + ( V 2 ( r ) − 1 + ξ 4 3 ) w + ( V 2 ( r ) − V 1 ( r ) ) Q 1 = 0 ,

where ξ lies between Q 1 and Q 2 . Combining the Sturm comparison theorem in [7] with the global energy inequality, if a 2 > a 1 and the energy density is strictly larger for all r , then Q 2 either changes sign at some finite r or fails to decay at infinity, both of which contradict that Q 2 is a positive radial decaying solution. Therefore, the assumption is invalid, and there do not exist two distinct positive radial decaying solutions. In conclusion, the equation has a unique positive radial solution Q in H 1 ( ℝ 3 ) , and Q ∈ Σ A ( ℝ 3 ) . The proof is complete.

Remark 1.2 At present, it remains unclear whether a blow-up solution exists for the mass-critical Equation (1.1) when the initial mass equals the minimal mass, i.e., ‖ ψ 0 ‖ 2 = ‖ Q ‖ 2 .

Next we derive a sharp threshold for the mass-supercritical case that separates global existence from finite-time blow-up.

Theorem 1.3 Let 4 3 < p < 4 . Let ψ 0 ∈ Σ A ( ℝ 3 ) be such that E 0 ( ψ 0 ) ≥ 0 and

(1.7) E 0 ( ψ 0 ) [ M ( ψ 0 ) ] œ c r < E 0 ( Q ) [ M ( Q ) ] œ c r ,

where E 0 is as in (1.4) and

(1.8) E 0 ( f ) : = 1 2 ‖ ∇ f ‖ L 2 2 + 1 4 ∫ ℝ 3 ( 1 | x | ∗ | f | 2 ) | f | 2 d x − 1 p + 2 ‖ f ‖ L p + 2 p + 2

1) If

(1.9) ‖ ∇ ψ 0 ‖ L 2 ‖ ψ 0 ‖ L 2 œ c r < ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r ,

then the corresponding solution to (1.1) exists globally in time, i.e., T * = ∞ and satisfies

‖ ∇ ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r < ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r ,

for all t ∈ [ 0 , ∞ ) .

2) If

(1.10) ‖ ∇ ψ 0 ‖ L 2 ‖ ψ 0 ‖ L 2 œ c r > ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r ,

then the corresponding solution to (1.1) satisfies

‖ ∇ ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r < ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r

for all t ∈ [ 0 , T * ) . Moreover, the solution blows up in finite time, i.e., T * < ∞ .

Remark 1.3 We only consider the case E 0 ( ψ 0 ) ≥ 0 , since according to Theorem 1.1, any solution with E 0 ( ψ 0 ) < 0 will blow up in finite time. Furthermore, it follows from (3.5) that there is no function ψ 0 ∈ Σ A ( ℝ 3 ) that satisfies both (1.9) and

‖ ∇ ψ 0 ‖ L 2 ‖ ψ 0 ‖ L 2 œ c r = ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r .

Here Theorem 1.3 indeed gives a sharp threshold for global existence versus finite time blow-up for (1.1).

Remark 1.4 In the case without a magnetic potential, Holmer and Roudenko have already proven this type of result. They further showed that as time t tends to infinity, the global solution scatters and asymptotically approaches the solution of the linear equation.

However, in the presence of a constant magnetic field, this scattering result is not expected to hold, because the Strichartz estimates associated with the magnetic Schrödinger operator are only available for finite times.

2. Preliminaries

In this section, we recall some basic properties of the magnetic Sobolev space H A 1 ( ℝ 3 ) and preliminary results that will be used later. Firstly, let us recall the local theory for the Cauchy problem (1.1).

Lemma 2.1 1) [5] Let N = 3 and 4 3 < p < 4 , then the following sharp Gagliardo-Nirenberg inequality

(2.1) ‖ ψ ‖ L p + 2 p + 2 ≤ C o p t ‖ ψ ‖ L 2 4 − p 2 ‖ ∇ ψ ‖ L 2 3 p / 2

holds for any ψ ∈ H A 1 ( ℝ 3 ) . The sharp constant C o p t is

C o p t = ( p + 2 ) 2 1 ‖ R ‖ L 2 p ,

where R is a radically ground state solution of the elliptic equation

(2.2) − Δ R + R − | R | P R = 0.

2) [22] Let N = 3 ,   4 3 < p < 4 , then

∫ ℝ 3 ( 1 | x | ∗ | ψ | 2 ) | ψ | 2 d x ≤ C G N ‖ ∇ ψ ‖ L 2 ‖ ψ ‖ L 2 3 .

The best constant C G N is defined by

C G N = 2 ‖ W ‖ L 2 2 ,

where W is a radically ground state solution of the elliptic equation

− Δ W + W = ( 1 | x | ∗ W 2 ) W .

Lemma 2.2 [14] Let A ∈ L l o c 2 ( ℝ 3 , ℝ 3 ) . Then H A 1 ( ℝ 3 ) equipped with the inner product

〈 f , g 〉 H A 1 : = ∫ ℝ 3 f   g ¯ d x + ∫ ℝ 3 ( ∇ + i A ) f ⋅ ( ∇ + i A ) g ¯   d x

is a Hilbert space.

Lemma 2.3 (Diamagnetic inequality [23]) Let A ∈ L l o c 2 ( ℝ 3 , ℝ 3 ) and f ∈ H A 1 ( ℝ 3 ) . then | f | ∈ H A 1 ( ℝ 3 ) in particular, we have

(2.3) | ∇ | f | ( x ) | < | ( ∇ + i A ) f ( x ) | ,     a . e .   x ∈ ℝ 3 .

3. Global Existence and Finite Time Blow-Up

In this section, we investigate the existence of solutions to (1.1) that are global in time as well as those exhibiting finite time blow-up. We begin with the subsequent virial identity associated with (1.1), which is of great significance in establishing the existence of finite time blow-up solutions.

Lemma 3.1 [24] Let 0 < p < 4 and ψ 0 ∈ Σ A ( ℝ 3 ) . Let ψ : [ 0 , T * ) × ℝ 3 → ℂ be the corresponding solution to (1.1). Set

(3.1) F ( ψ ( t ) ) : = ∫ ℝ 3 | x | 2 | ψ ( t , x ) | 2 d x .

Then the function [ 0 , T * ) ∋ t ↦ F ( ψ ( t ) ) is in C 2 ( [ 0 , T * ) ) and

F ′ ( ψ ( t ) ) = 4 Im ∫ ℝ 3 x ⋅ ∇ ψ ( t , x ) ψ ¯ ( t , x ) d x , F ″ ( ψ ( t ) ) = 16 E ( ψ ( t ) ) − 12 p − 6 p + 2 ‖ ψ ( t ) ‖ p + 2 p + 2 − 2 ∫ ℝ 3 ( 1 | x | ∗ | ψ ( t ) | 2 ) | ψ ( t ) | 2 d x

for all t ∈ [ 0 , T * ) .

Now we prove the sharp threshold for global existence versus blow-up for (1.1) in the mass-critical case given in Proposition 1.2.

Proof of Theorem1.2. 1) By the Gagliardo-Nirenberg inequality and the diamagnetic inequality (2.3), we have

(3.2) ‖ f ‖ L 10 3 10 3 ≤ 5 3 ( ‖ f ‖ L 2 ‖ R ‖ L 2 ) 4 3 ‖ ∇ f ‖ L 2 2 ≤ 5 3 ( ‖ f ‖ L 2 ‖ R ‖ L 2 ) 4 3 ‖ ( ∇ + i A ) f ‖ L 2 2 ,

where R is a unique positive radial solution to (2.2) with p = 4 3 . From this inequality and the conservation laws of mass and energy, we infer that

E ( ψ 0 ) = E ( ψ ( t ) ) = 1 2 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 2 + 1 4 ∫ ℝ 3 ( 1 | x | ∗ | ψ ( t ) | 2 ) | ψ ( t ) | 2 d x − 3 10 ‖ ψ ( t ) ‖ L 10 3 10 3 ≥ 1 2 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 2 − 3 10 ‖ ψ ( t ) ‖ L 10 3 10 3 ≥ 1 2 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 2 − 1 2 ( ‖ ψ ( t ) ‖ L 2 ‖ Q ‖ L 2 ) 4 3 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 2 = 1 2 ( 1 − ( ‖ ψ ( t ) ‖ L 2 ‖ Q ‖ L 2 ) 4 3 ) ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 2

for all t ∈ [ 0 , T * ) . As ‖ ψ 0 ‖ L 2 < ‖ Q ‖ L 2 , we have sup t ∈ [ 0 , T * ) ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 ≤ C , which by the blow-up alternative, implies that T * = ∞ .

2) Let c > ‖ Q ‖ L 2 , then c > ‖ Q ‖ L 2 . We define

ψ 0 ( x ) : = a λ 3 2 Q ( λ x ) ,

where a : = c ‖ Q ‖ L 2 > 1 and the parameter λ > 0 will be chosen subsequently. Give that Q decays exponentially at infinity, it is evident that Q ∈ Σ A ( ℝ 3 ) . Furthermore, we get

‖ ψ 0 ‖ L 2 2 = a 2 ‖ Q ‖ L 2 2 ,     ‖ ∇ ψ 0 ‖ L 2 2 = a 2 λ 2 ‖ ∇ Q ‖ L 2 2 , ‖ ψ 0 ‖ L 10 3 10 3 = a 10 3 λ 2 ‖ Q ‖ L 10 3 10 3 ,     ‖ ρ ψ 0 ‖ L 2 2 = a 2 λ − 2 ‖ ρ Q ‖ L 2 2 , ∫ ℝ 3 ( 1 | x | ∗ | ψ 0 | 2 ) | ψ 0 | 2 d x = a 4 λ 4 ∫ ℝ 3 ( 1 | x | ∗ | Q | 2 ) | Q | 2 d x .

It follows that

E 0 ( ψ 0 ) = 1 2 ‖ ∇ ψ 0 ‖ L 2 2 + b 2 8 ‖ ρ ψ 0 ‖ L 2 2 + 1 4 ∫ ℝ 3 ( 1 | x | ∗ | ψ 0 | 2 ) | ψ 0 | 2 d x − 3 10 ‖ ψ 0 ‖ L 10 3 10 3 = 1 2 a 2 λ 2 ‖ ∇ Q ‖ L 2 2 + b 2 8 a 2 λ − 2 ‖ ρ Q ‖ L 2 2 + 1 4 a 4 λ 4 ∫ ℝ 3 ( 1 | x | ∗ | Q | 2 ) | Q | 2 d x       − 3 10 a 10 3 λ 2 ‖ Q ‖ L 10 3 10 3 = a 2 λ 2 ( 1 2 ‖ ∇ Q ‖ L 2 2 + b 2 8 λ − 4 ‖ ρ Q ‖ L 2 2 + 1 4 a 2 λ 2 ∫ ℝ 3 ( 1 | x | ∗ | Q | 2 ) | Q | 2 d x       − 3 10 a 10 3 ‖ Q ‖ L 10 3 10 3 ) .

Using Pohozaev’s identity (see [21])

− 1 2 ‖ ∇ Q ‖ L 2 2 + 3 2 ‖ Q ‖ L 2 2 + 3 4 ∫ ℝ 3 ( 1 | x | ∗ | Q | 2 ) | Q | 2 d x + 3 p + 2 ‖ Q ‖ L p + 2 p + 2 = 0 ,

then

E 0 ( ψ 0 ) = a 2 λ 2 ( − ( 1 2 + a 2 λ 2 6 ) ‖ ∇ Q ‖ L 2 2 + b 2 8 λ − 4 ‖ ρ Q ‖ L 2 2 − 1 2 ‖ Q ‖ L 2 2       − 3 10 ( a 2 λ 2 + a 10 3 ) ‖ Q ‖ L 10 3 10 3 ) .

Taking λ > 0 sufficiently large, we have E 0 ( ψ 0 ) < 0 . According to Theorem1.1 the solution corresponding to Equation (1.1), with initial condition ψ | t = 0 = ψ 0 exhibits finite-time blow-up. The proof is complete.

Proof of Theorem1.3. 1) Let us consider an initial condition ψ 0 ∈ Σ A ( ℝ 3 ) that fulfills conditions (1.7) and (1.9). Suppose ψ : [ 0 , T * ) × ℝ 3 → ℂ represents the corresponding solution to Equation (1.1). According to the Gagliardo-Nirenberg inequality, we obtain

E 0 ( ψ ( t ) ) [ M ( ψ ( t ) ) ] œ c r ≥ 1 2 ( ‖ ( ∇ + i A ) ( ψ ( t ) ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r ) 2 − C G N 4 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 3 + 2 oe c r       + b 2 8 ‖ ρ ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 2 oe c r − C o p t p + 2 ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 3 p 2 ‖ ψ ( t ) ‖ L 2 4 − p 2 + 2 oe c r ≥ G ( ‖ ∇ ( ψ ( t ) ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r )

for all t ∈ [ 0 , T * ) . Using (1.5) and (1.7), we have

(3.3) G ( ‖ ∇ ( ψ ( t ) ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r ) ≤ E 0 ( ψ 0 ) [ M ( ψ 0 ) ] œ c r < E 0 ( Q ) [ M ( Q ) ] œ c r ≤ G ( ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r )

for all t ∈ [ 0 , T * ) . By (1.9), the continuity argument implies

‖ ∇ ( ψ ( t ) ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r < ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r

for all t ∈ [ 0 , T * ) . By the conservation of mass, we infer that

sup t ∈ [ 0 , T * ) ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 ≤ C ( ‖ ψ ‖ L 2 , ‖ Q ‖ L 2 , ‖ ∇ Q ‖ L 2 )

on the flip side, utilizing (1.5) and Gagliardo-Nirenberg we obtain

b 2 8 ‖ ρ ψ ( t ) ‖ L 2 2 = E 0 ( ψ ( t ) ) − 1 2 ‖ ∇ ψ ( t ) ‖ L 2 2 − 1 4 ∫ ℝ 3 ( 1 | x | ∗ | ψ ( t ) | 2 ) | ψ ( t ) | 2 d x       + 1 p + 2 ‖ ψ ( t ) ‖ L p + 2 p + 2 ≤ E 0 ( ψ ( t ) ) − 1 4 ∫ ℝ 3 ( 1 | x | ∗ | ψ ( t ) | 2 ) | ψ ( t ) | 2 d x + 1 p + 2 ‖ ψ ( t ) ‖ L p + 2 p + 2 ≤ E 0 ( ψ 0 ) − C G N 4 ‖ ∇ ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 3 + C o p t p + 2 ‖ ∇ ψ ( t ) ‖ L 2 3 p 2 ‖ ψ ( t ) ‖ L 2 4 − p 2 ≤ C ( E 0 ( ψ 0 ) , M ( ψ 0 ) , ‖ ∇ Q ‖ L 2 , ‖ Q ‖ L 2 )

for all t ∈ [ 0 , T * ) . By (1.3) and the conservation of angular momentum, we have

sup t ∈ [ 0 , T * ) ‖ ( ∇ + i A ) ψ ( t ) ‖ L 2 ≤ C ( E 0 ( ψ 0 ) , M ( ψ 0 ) , ‖ ∇ Q ‖ L 2 , ‖ Q ‖ L 2 ) ,

which by the blow-up alternative, implies that T * = ∞ .

2) Let us now examine ψ 0 ∈ Σ A ( ℝ 3 ) , which satisfies both conditions (1.7) and (1.10).

By employing the identical logical approach as used before, we can deduce that

(3.4) ‖ ∇ ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r > ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r

for all t ∈ [ 0 , T * ) . Next, we will demonstrate that the solution experiences blow-up within a finite timeframe. By drawing from Equation (1.7), we select a parameter ϑ = ϑ ( ψ 0 , Q ) > 0 , such that

E ( ψ 0 ) [ M ( ψ 0 ) ] œ c r ≤ ( 1 − ϑ ) E 0 ( Q ) [ M ( Q ) ] œ c r .

We also denote

(3.5) H ( f ) : = ‖ ∇ f ‖ L 2 2 + b 2 4 ‖ ρ f ‖ L 2 2 + 1 4 ∫ ℝ 3 ( 1 | x | ∗ | f | 2 ) | f | 2 d x − 3 p 2 ( p + 2 ) ‖ f ‖ L p + 2 p + 2 = 3 p 2 E 0 ( f ) − 3 p − 4 4 ‖ ∇ f ‖ L 2 2 − 3 p − 2 8 ∫ ℝ 3 ( 1 | x | ∗ | f | 2 ) | f | 2 d x       − ( 3 p − 4 ) b 2 16 ‖ ρ f ‖ L 2 2

From (1.5) (3.6), the conservation of mass, we see that

H ( ψ ( t ) ) [ M ( ψ ( t ) ) ] œ c r ≤ 3 p 2 E ( ψ ( t ) ) [ M ( ψ ( t ) ) ] œ c r − 3 p − 4 4 ( ‖ ∇ ψ ( t ) ‖ L 2 ‖ ψ ( t ) ‖ L 2 œ c r ) 2 ≤ 3 p 2 E ( ψ ( t ) ) [ M ( ψ ( t ) ) ] œ c r − 3 p − 4 4 ( ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r ) 2 ≤ 3 p 2 ( 1 − ϑ ) E 0 ( Q ) [ M ( Q ) ] œ c r − 3 p − 4 4 ( ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r ) 2 = − 3 p − 4 4 ϑ ( ‖ ∇ Q ‖ L 2 ‖ Q ‖ L 2 œ c r ) 2

for all t ∈ [ 0 , T * ) . The following can be derived from Lemma 3.1.

F ″ ( ψ ( t ) ) = 8 H ( ψ ( t ) ) ≤ − 2 ( 3 p − 4 ) ϑ ( ‖ Q ‖ L 2 ‖ ψ 0 ‖ L 2 ) 2 oe c r ‖ ∇ Q ‖ L 2 2 < 0

for all t ∈ [ 0 , T * ) . This proves that T * < ∞ . The proof is complete.

4. Conclusion

This paper studies the nonlinear Schrödinger equation with a constant magnetic field, and identifies the precise thresholds for the global existence and finite-time blow-up of its solutions in the mass-critical and mass-supercritical regimes where 4 3 < p < 4 . In the mass-critical case, the L 2 norm of Q the unique positiveradial solution to a certain elliptic equation serves as the mass threshold, solutions exist globally for initial data with mass below this threshold, while there exist initial data with mass above it that lead to finite-time blow-up, and the case of critical initial mass remains an open problem. For the mass-supercritical case, double thresholds based on the energy-mass and gradient-mass products associated with Q are established. Solutions exist globally when specific inequalities for these products are satisfied, and finite-time blow-up occurs when the inequalities are reversed. Meanwhile, several sufficient conditions for the finite-time blow-up of solutions are derived. In addition, the presence of a magnetic field makes the solutions of the equation lack scattering properties, whereas the scattering conclusion for the equation without a magnetic field has been proven.

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