Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 24, pp. 1–13.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

SMALL DATA BLOW-UP OF SOLUTIONS TO NONLINEAR

SCHRÖDINGER EQUATIONS WITHOUT

GAUGE INVARIANCE IN L2

YUANYUAN REN, YONGSHENG LI

Communicated by Jesus Ildefonso Diaz

Abstract. In this article we study the Cauchy problem of the nonlinear

Schrödinger equations without gauge invariance

i∂tu+ ∆u = λ(|u|p1 + |v|p2 ), (t, x) ∈ [0, T )× Rn,

i∂tv + ∆v = λ(|u|p2 + |v|p1 ), (t, x) ∈ [0, T )× Rn,

where 1 < p1, p2 < 1 + 4/n and λ ∈ C\{0}. We first prove the existence of

a local solution with initial data in L2(Rn). Then under a suitable condition

on the initial data, we show that the L2-norm of the solution must blow up in
finite time although the initial data are arbitrarily small. As a by-product, we

also obtain an upper bound of the maximal existence time of the solution.

1. Introduction

In this article, we consider the Cauchy problem of the nonlinear Schrödinger
equations without gauge invariance

i∂tu+ ∆u = λ(|u|p1 + |v|p2), (t, x) ∈ [0, T )× Rn,
i∂tv + ∆v = λ(|u|p2 + |v|p1), (t, x) ∈ [0, T )× Rn,

u(0, x) = εf(x), x ∈ Rn,
v(0, x) = εg(x), x ∈ Rn,

(1.1)

where 1 < p1, p2 < 1 + 4
n , T > 0 and ε > 0 is a small parameter. u = u(t, x) and

v = v(t, x) are complex-valued unknown functions, f = f1 + if2 and g = g1 + ig2

are prescribed complex-valued functions, and λ = λ1 + iλ2 ∈ C\{0}.
System (1.1) is a generalization of the Cauchy problem of the nonlinear equation

i∂tu+ ∆u = F (u), (t, x) ∈ [0, T )× Rn,
u(0, x) = f(x), x ∈ Rn,

(1.2)

where F (u) = λ|u|p.
We know that a solution to (1.2) on [0, T ] gives rise to a family of solutions, i.e.

for any γ > 0,

uγ(t, x) := γ
2
p−1u(γ2t, γx)

2010 Mathematics Subject Classification. 35Q55, 35B44.

Key words and phrases. Nonlinear Schrödinger equations; weak solution; blow up of solutions.
c©2021 Texas State University.

Submitted February 15, 2018. Published March 31, 2021.

1



2 Y. REN, Y. LI EJDE-2021/24

is also a solution to (1.2) on [0, T/γ2]. Moreover, a direct calculation gives

‖uγ(t, ·)‖L2(Rn) = γ
2
p−1−

n
2 ‖u(t, ·)‖L2(Rn).

Thus if the order p satisfies

2

p− 1
− n

2
= 0 i.e. p = p0 := 1 +

4

n
,

then the L2-norm of the solution is also scale invariant. Therefore, the case p = p0

is called L2-critical case. The case of p < p0 (resp. p > p0) is called L2-subcritical
case (resp. L2-supercritical case).

We say that a nonlinear function F satisfies the gauge invariance if F (eiθu) =
eiθF (u) for θ ∈ R. However, the nonlinear term in (1.2) F (u) = λ|u|p is not
gauge invariant. This is different from F (u) = λ|u|p−1u, which satisfies the gauge
invariance and possesses the conservation of mass (and also energy for H1-solution).
However, in the case of non-gauge invariance, the conservation of mass (or energy
for H1-solution) fails (see [9]).

Equation (1.2) has various physical contexts and has been studied from the
mathematical viewpoint in several papers. For example, it is related to the Gross-
Pitaevskii equation, which describes the Bose-Einstein condensate in physics. The
solution Φ of the Gross-Pitaevskii equation satisfies a non-zero constant boundary
condition as |x| tends to infinity. In that case, the nonlinearity |u|p appears if
we introduce the new dynamical variable u by Φ = u + constant and expand the
nonlinearity |Φ|pΦ in u (see [8, 17]). Thus, it is expected that the analysis of (1.2)
may be helpful for the study of the Gross-Pitaevskii equation.

For (1.2), in the single equation case, when 1 < p < 1 + 4
n−2s (0 ≤ s < n

2 ), it is

well known that local well-posedness holds in Sobolev spaces Hs (see [3, 21] with
the references therein). In one dimension, when p = 2, Kenig et al. [14] first proved
the local well-posedness in Hs(R) when s > − 1

4 . For general dimension, when p is

sufficiently large, the small initial data L2-solution exists globally. More precisely,

for L2 ∩ L1+ 1
p -data, when pS < p < p0 = 1 + 4

n , where pS = n+2+
√
n2+4n+12
2n is the

Strauss exponent (see [18]), which is greater than 1 + 2
n and less than 1 + 4

n , the

global existence result for small initial data holds (see also [3]). When 1 < p ≤ 1+ 2
n ,

Ikeda and Wakasugi [11] showed that the L2-norm of the solution for (1.2) blows
up at finite time, provided that

λ1 Im

∫
Rn
f(x) dx < 0, or λ2 · Re

∫
Rn
f(x) dx > 0.

In particular, this implies that there is no global well-posedness even for small
initial data. Later, in [9] Ikeda and Inui proved a small initial data blow-up result
of the L2-solution for (1.2) in 1 < p < p0. Recently, Ikeda and Inui [10] proved the
non-existence of the local weak-solution for (1.2) in the L2-supercritical case p > p0

for suitable L2-data. To construct the blow up solution, the authors in [11, 9, 10]
used a test-function method which heavily relies on the shape of the initial data,
though their norms may be arbitrarily small.

The coupled nonlinear Schrödinger equations

i∂tu+ ∆u = λ(|u|p1 + |v|p2)u, (t, x) ∈ [0, T )× Rn,
i∂tv + ∆v = λ(|v|p1 + |u|p2)v, (t, x) ∈ [0, T )× Rn,

(1.3)



EJDE-2021/24 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 3

where 1 < p1, p2 < 1 + 4
n , describe the minimum approximation of the transforma-

tion of light wave. For more details of the physical background, we refer the readers
to [2, 1, 16, 20]. When u and v satisfy non-zero constant boundary condition as |x|
tends to infinity, the same analysis as for (1.2) leads to (1.1).

In this article, our main aim is to prove a small initial data blow-up result of
L2-solution for (1.1) in the subcritical case 1 < p1, p2 < p0. We also obtain an
upper bound of the lifespan for (1.1) when 1 < p1, p2 < p0.

For the rest of this article, we let p := min{p1, p2}. Since 1 < p1, p2 < p0, we
have p ∈ (1, p0). We impose the additional assumption on the initial data,

λ2(f1(x) + g1(x)) ≥

{
|x|−k, if |x| ≥ 1,

0, if |x| < 1,
or

−λ1(f2(x) + g2(x)) ≥

{
|x|−k, if |x| ≥ 1,

0, if |x| < 1,

(1.4)

where n/2 < k < 2/(p− 1). Note that such k exists if and only if 1 < p < p0. Now,
we can state our main result.

Theorem 1.1. Let 1 < p1, p2 < 1+ 4
n , λ = λ1 + iλ2 ∈ C\{0} and f, g ∈ L2(Rn). If

f and g satisfy initial data condition (1.4), then there exist ε0 > 0 and a constant
C = C(k, p1, p2, λ) > 0 such that for any ε ∈ (0, ε0),

Tε ≤ Cε−1/θ,

where θ = 1
p−1 −

k
2 . Moreover, the L2-norm of the local solution blows up in finite

time,

lim
t→T−ε

(‖u(t)‖L2 + ‖v(t)‖L2) =∞. (1.5)

The definition of Tε can be found in (2.6) below. This theorem gives an upper
bound of the local existence time to the Cauchy problem (1.1) in L2(Rn). At the
same time, we note that (1.5) means that the conservation law of mass does not
hold for equation (1.1).

The rest of this paper is arranged as follows. In Section 2, we prove the local
well-posedness for (1.1) with initial data in L2(Rn) and give the definition of L2-
solution. In Section 3, we show that an L2-solution of (2.1) on [0, T ) is a weak
solution of (1.1). In Section 4, we give the proof of Theorem 1.1.

We concluding this section, by introducing some notation. For 1 ≤ r ≤ ∞,
let Lr = Lr(Rn) denote the usual Lebesgue space. For a time interval I, we
use a time-space Lebesgue space Lq(I;Lr(Rn)), with the norm ‖u‖Lq(I;Lr(Rn)) :=
‖‖u(t)‖Lr(Rn)‖Lq(I). We often omit the time interval I and Rn and denote simply
Lq(I;Lr(Rn)) as LqLr, when no confusion may occur. We write A . B if there
exists a constant C > 0 such that A ≤ CB.

2. Local well-posedness

Firstly, by the Duhamel formula, we consider the integral equations

u(t) = εS(t)f − iλ
∫ t

0

S(t− τ) (|u|p1 + |v|p2) dτ

v(t) = εS(t)g − iλ
∫ t

0

S(t− τ)(|u|p2 + |v|p1) dτ,

(2.1)



4 Y. REN, Y. LI EJDE-2021/24

as the integral version of the Cauchy problem (1.1), where S(t) = eit∆ is the free
evolution group of the linear Schrödinger equation in Hs(Rn).

Definition 2.1 ([3, 19]). The pair (q, r) of real numbers is said to be admissible if
2
q = n

2 −
n
r and

2 ≤ r < 2n

n− 2
(2 ≤ r ≤ ∞ if n = 1; 2 ≤ r <∞ if n = 2).

Next, we define the function space

XT = C([0, T );L2(Rn)) ∩ Lq1((0, T );Lr1(Rn)) ∩ Lq2((0, T );Lr2(Rn)),

where (qj , rj) is an admissible pair defined by rj = pj + 1, j = 1, 2.

Lemma 2.2 ([3, 19]). Let (q, r) and (γ, ρ) be any admissible pairs. For any time
interval I, we have the estimates

‖S(·)ϕ‖Lq(R,Lr(Rn)) ≤ C‖ϕ‖L2 ,

‖
∫ t

0

S(t− s)F (s) ds‖Lq(I,Lr(Rn)) ≤ C‖F‖Lγ′ (I,Lρ′ (Rn)) .

Theorem 2.3. Let 1 < p1, p2 < 1+ 4
n , λ ∈ C, ε > 0 and f, g ∈ L2(Rn). Then there

exist a positive time T = T (ε, ‖f‖L2 , ‖g‖L2) and a unique solution (u, v) ∈ XT×XT

of (2.1).

The proof of this theorem is based on contractive mapping principle. See [21, 4,
13, 7] for the gauge invariance case. For the convenience of the reader, we give a
brief proof.

Proof. Let R > 0 and B(R) = {(u, v)|u, v ∈ XT , ‖u‖XT ≤ R, ‖v‖XT ≤ R}, where

‖u‖XT = ‖u‖L∞L2 + ‖u‖Lq1Lr1 + ‖u‖Lq2Lr2 . (2.2)

Endowed with the metric

d((u1, v1), (u2, v2)) = ‖u1 − u2‖XT + ‖v1 − v2‖XT ,
It is easy to see that, B(R) is a complete metric space.

We expect to find the proper conditions of T and R, which imply that Γ : (u, v) 7→
(Γ1u,Γ2v), given by

Γ1u(t) = εS(t)f − iλ
∫ t

0

S(t− τ)(|u|p1 + |v|p2) dτ

Γ2v(t) = εS(t)g − iλ
∫ t

0

S(t− τ)(|u|p2 + |v|p1) dτ,

is a strict contraction on B(R).
For (u1, v1), (u2, v2) ∈ B(R), we have

‖Γ1u1 − Γ1u2‖XT

≤ |λ|‖
∫ t

0

S(t− τ)(|u1|p1 − |u2|p1) dτ‖XT + |λ|‖
∫ t

0

S(t− τ)(|v1|p2 − |v2|p2) dτ‖XT

:= I + II.

By Lemma 2.2 and Hölder’s inequality, we obtain

I .‖ |u1|p1 − |u2|p1‖Lq′1Lr′1



EJDE-2021/24 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 5

.Tα
(
‖u1‖p1−1

Lq1Lr1 + ‖u2‖p1−1
Lq1Lr1

)
‖u1 − u2‖Lq1Lr1

.TαRp1−1‖u1 − u2‖Lq1Lr1 ,

where 1
r′1

= p1
p1+1 , 1

q′1
= p1

q1
+ α, α = n

4 (1 + 4
n − p1) > 0, and

II . T β(‖v1‖p2−1
Lq2Lr2 + ‖v2‖p2−1

Lq2Lr2 )‖v1 − v2‖Lq2Lr2 . T βRp2−1‖v1 − v2‖Lq2Lr2 ,

where 1
r′2

= p2
p2+1 , 1

q′2
= p2

q2
+ β, β = n

4 (1 + 4
n − p2) > 0. So, we have

‖Γ1u1 − Γ1u2‖XT . TαRp1−1‖u1 − u2‖Lq1Lr1 + T βRp2−1‖v1 − v2‖Lq2Lr2 . (2.3)

Similarly, we obtain

‖Γ2v1 − Γ2v2‖XT . TαRp1−1‖v1 − v2‖Lq1Lr1 + T βRp2−1‖u1 − u2‖Lq2Lr2 . (2.4)

Combining (2.3) with (2.4), we have

d
(
Γ(u1, v1),Γ(u2, v2)

)
=‖Γ1u1 − Γ1u2‖XT + ‖Γ2v1 − Γ2v2‖XT
.TαRp1−1(‖u1 − u2‖Lq1Lr1 + ‖v1 − v2‖Lq1Lr1 )

+ T βRp2−1(‖u1 − u2‖Lq2Lr2 + ‖v1 − v2‖Lq2Lr2 ),

Let

T ≤ min{(4Rp1−1)−1/α, (4Rp2−1)−1/β}. (2.5)

Then there exists a constant δ ∈ (0, 1) such that

d(Γ(u1, v1),Γ(u2, v2)) < δ(‖u1 − u2‖XT + ‖v1 − v2‖XT ).

It follows that Γ is a strict contraction on B(R), and thus has a unique fixed point
(u, v). This completes the proof. �

The above solution (u, v) is called an “L2-solution”. Let Tε be the maximal
existence time of the local L2-solution,

Tε = sup
{
T ∈ (0,∞] : a unique solution (u, v) to (2.1) exists and

belongs to XT ×XT

}
.

(2.6)

Then (2.5) provides lower bound of lifespan.

Corollary 2.4. Under the the assumptionsin Theorem 2.3, we have the estimate

Tε ≥ C min(ε−1/θ1 , ε−1/θ2),

where θj = 1
pj−1 −

n
4 > 0, j = 1, 2 and C = C(n, p1, p2, ‖f‖L2 , ‖g‖L2) > 0 is a

constant.

Combining Theorem 1.1 with Corollary 2.4, we obtain the estimate of the lifespan

min(ε−1/θ1 , ε−1/θ2) . Tε . ε
−1/θ.

However, it is not optimal. Actually, to the best of our knowledge, if p1 < p2, we
have p = min{p1, p2} = p1, then the following estimate holds for sufficiently small
ε > 0,

ε−1/θ1 . Tε . ε
−1/θ.

But we know that

θ − θ1 =
n

4
− k

2
< 0.



6 Y. REN, Y. LI EJDE-2021/24

Similarly, if p2 < p1, then ε−1/θ2 . Tε . ε−1/θ holds. However, this is also not
optimal. For the time being, to our knowledge, the optimal order of the lifespan is
an open question.

3. Weak solutions

To obtain our main results, we first define a weak solution of (1.1).

Definition 3.1. Let T > 0. (u, v) is a weak solution of (1.1) on [0, T ), if (u, v) ∈
Lp1loc([0, T )× Rn) ∩ Lp2loc([0, T )× Rn) and satisfies∫

[0,T )×Rn
u(−i∂tψ + ∆ψ) dx dt

= iε

∫
Rn
f(x)ψ(0, x) dx+ λ

∫
[0,T )×Rn

(|u|p1 + |v|p2)ψ dx dt,

(3.1)

∫
[0,T )×Rn

v(−i∂tψ + ∆ψ) dx dt

= iε

∫
Rn
g(x)ψ(0, x) dx+ λ

∫
[0,T )×Rn

(|u|p2 + |v|p1)ψ dx dt

(3.2)

for any ψ ∈ C2
0 ([0, T )×Rn). Moreover, if T can be chosen arbitrary large, then we

say that (u, v) is a global weak solution of (1.1).

We note that an L2-solution as in Theorem 2.3 is always a weak solution in the
sense of Definition 3.1. Then, we have the following proposition.

Proposition 3.2. Let T > 0. If (u, v) is an L2-solution of (2.1) on [0, T ), then
(u, v) is also a weak solution on [0, T ) in the sense of Definition 3.1.

Proof. Let T > 0 and (qj , rj) be admissible pairs, where rj = pj + 1, j = 1, 2. Let
(u, v) be an L2-solution to (2.1) on [0, T ) and ψ ∈ C2

0 ([0, T )×Rn). It is easy to see
that

u, v ∈ Lp1loc([0, T )× Rn) ∩ Lp2loc([0, T )× Rn).

Let u = U1 + U2, where

U1 = εS(t)f, U2 = −iλ
∫ t

0

S(t− τ)(|u|p1 + |v|p2) dτ.

By a standard density argument and integration by parts, we can obtain, for any
ψ ∈ C2

0 ([0, T )× Rn),∫
[0,T )×Rn

U1(−i∂tψ + ∆ψ) dx dt = i

∫
Rn
εf(x)ψ(0, x) dx.

Thus, it suffices to prove that∫
[0,T )×Rn

U2(−i∂tψ + ∆ψ) dx dt = λ

∫
[0,T )×Rn

(|u|p1 + |v|p2)ψ dx dt. (3.3)

Let

K1 =

∫
[0,T )×Rn

U2∆ψ dx dt, K2 = −i
∫

[0,T )×Rn
U2∂tψ dx dt,

K = λ

∫
[0,T )×Rn

(|u|p1 + |v|p2)ψ dx dt.

(3.4)

So, it is sufficiently to prove that K = K1 +K2.



EJDE-2021/24 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 7

Since u, v ∈ Lq1Lr1 ∩ Lq2Lr2 , and C∞0 ([0, T )× Rn) is dense in Lq1Lr1 ∩ Lq2Lr2 ,
there exist two sequences {uk}k∈N, {vk}k∈N in C∞0 ([0, T )× Rn), such that

lim
k→∞

‖uk − u‖Lq1Lr1∩Lq2Lr2 = 0, lim
k→∞

‖vk − v‖Lq1Lr1∩Lq2Lr2 = 0.

We also introduce an approximate sequence {U2,k}k∈N to U2,

U2,k = −iλ
∫ t

0

S(t− τ)(|uk|p1 + |vk|p2) dτ.

By Lemma 2.2 and Hölder’s inequality with 1
r′j

= 1
rj

+
pj−1
pj+1 and 1

q′j
= 1

qj
+
pj−1
qj

+αj ,

where αj = n
4 (1 + 4

n − pj) > 0, j = 1, 2, we obtain

‖U2 − U2,k‖L∞L2

. ‖
∫ t

0

S(t− τ)(|u|p1 − |uk|p1) dτ‖L∞L2 + ‖
∫ t

0

S(t− τ)(|v|p2 − |vk|p2) dτ‖L∞L2

. ‖(|u|p1 − |uk|p1)‖
Lq
′
1Lr
′
1

+ ‖(|v|p2 − |vk|p2)‖
Lq
′
2Lr
′
2

. Tα1‖u− uk‖Lq1Lr1 (‖u‖p1−1
Lq1Lr1 + ‖uk‖p1−1

Lq1Lr1 )

+ Tα2‖v − vk‖Lq2Lr2 (‖v‖p2−1
Lq2Lr2 + ‖vk‖p2−1

Lq2Lr2 ).

(3.5)
Noting U2,k(0, x) = 0, by (3.5) and integration by parts, we have

K2 = −i lim
k→∞

∫
[0,T )×Rn

U2,k∂tψ dx dt = i lim
k→∞

∫
[0,T )×Rn

∂tU2,kψ dx dt. (3.6)

By almost the same argument as in (3.5), we find that U2,k ∈ C([0, T );H1) and
the time derivative ∂tU2,k ∈ C([0, T );H−1) satisfy

i∂tU2,k + ∆U2,k = λ(|uk|p1 + |vk|p2). (3.7)

By changing variables with t− τ = τ ′, we have

∂tU2,k = −i∂t
∫ t

0

λS(t− τ)(|uk|p1 + |vk|p2)(τ) dτ

= −i∂t
∫ t

0

λS(τ ′)(|uk|p1 + |vk|p2)(t− τ ′) dτ ′

= −i
∫ t

0

λS(t− τ)∂t(|uk|p1 + |vk|p2)(τ) dτ − iλS(t)(|uk|p1 + |vk|p2)(0).

Applying Lemma 2.2, we have

‖∂tU2,k‖L2 . ‖∂t(|uk|p1)‖
Lq
′
1Lr
′
1

+ ‖∂t(|vk|p2)‖
Lq
′
2Lr
′
2

+ ‖uk(0)‖p1
L2p1

+ ‖vk(0)‖p2
L2p2

. Tα1‖uk‖p1−1
Lq1Lr1‖∂tuk‖Lq1Lr1 + Tα2‖vk‖p2−1

Lq2Lr2‖∂tvk‖Lq2Lr2
+ ‖uk‖p1L∞L2p1

+ ‖vk‖p1L∞L2p1
< +∞,

for any k ∈ N. Thus we obtain ∂tU2,k ∈ C([0, T );L2). Therefore from the identity
(3.7), we can find U2,k ∈ C([0, T );H2). Then we have

(∆U2,k, ψ)L2 = (U2,k, ∆ψ)L2 , ∀k ∈ N. (3.8)



8 Y. REN, Y. LI EJDE-2021/24

By the same way as for (3.5), we obtain∣∣∣ ∫
[0,T )×Rn

λ(|u|p1 + |v|p2)ψ dx dt−
∫

[0,T )×Rn
λ(|uk|p1 + |vk|p2)ψ dx dt

∣∣∣
.
∣∣ ∫

[0,T )×Rn
(|u|p1 − |uk|p1)ψ dx dt

∣∣+
∣∣ ∫

[0,T )×Rn
(|v|p2 − |vk|p2)ψ dx dt

∣∣
. Tα1‖u− uk‖Lq1Lr1

(
‖u‖p1−1

Lq1Lr1 + ‖uk‖p1−1
Lq1Lr1

)
‖ψ‖Lq1Lr1

+ Tα2‖v − vk‖Lq2Lr2
(
‖v‖p2−1

Lq2Lr2 + ‖vk‖p2−1
Lq2Lr2

)
‖ψ‖Lq2Lr2 ,

(3.9)

and ∣∣ ∫
[0,T )×Rn

(U2,k − U2)∆ψ
∣∣ . T‖U2,k − U2‖L∞L2‖∆ψ‖L∞L2 . (3.10)

Thus, combining (3.6)-(3.7) with (3.8)-(3.10), we obtain

K2 = lim
k→∞

(∫
[0,T )×Rn

λ(|uk|p1 + |vk|p2)ψ dx dt−
∫

[0,T )×Rn
ψ∆U2,k dx dt

)
= K − lim

k→∞

∫
[0,T )×Rn

U2,k∆ψ dx dt

= K −K1.

(3.11)

Combining (3.4) with (3.11), we obtain (3.3), thus (3.1) is valid. Similarly, (3.2) is
also valid. The proof is complete. �

4. Proof of main result

We first obtain an upper bound of lifespan via a test function method, inspired
by [15, 9]. For 1 < p1, p2 < 1 + 4

n , to use this method, we take the intermediate
variable p = min{p1, p2}. Then we give the proof of Theorem 1.1. Without loss of
generality, we assume that λ1 > 0. The other cases in (1.4) can be treated in the
almost same way.

We introduce the non-negative smooth radial bump function φ ∈ C2
0 (Rn) as

follows (see [5, 6, 9]),

φ(0) = 1, 0 < φ(x) ≤ 1, for |x| > 0,

where φ(x) is decreasing with respect to |x| and φ(x) → 0 as |x| → ∞ sufficiently
fast. Moreover, there exists µ > 0 such that

|∆φ| ≤ µφ, x ∈ Rn, (4.1)

and ‖φ‖L1 = 1. For sufficiently large θ, we set

η(t) =

{
(1− t/T )θ, if 0 ≤ t ≤ T,
0, if t > T,

where T > 0. Furthermore, for R > 0, we set

ηR(t) = η(t/R2), φR(x) = φ(x/R), ψR(t, x) = ηR(t)φR(x).

Next, we introduce some notation. Let Tε be the maximal existence time. For
T,R > 0 with TR2 < Tε, define

I1
R(T ) =

∫
[0,TR2)×Rn

(|u|p1 + |v|p2)ψR(t, x) dx dt,



EJDE-2021/24 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 9

I2
R(T ) =

∫
[0,TR2)×Rn

(|u|p2 + |v|p1)ψR(t, x) dx dt,

JR = −ε
∫
Rn

(f2(x) + g2(x))φR(x) dx,

H1(T ) = µ
(∫

[0,T )×Rn
η(t)φ(x) dx dt

)1/q

,

H2(T ) =
(∫

[0,T )×Rn
|∂tη(t)|qη(t)−q/pφ(x) dx dt

)1/q

,

where p = min{p1, p2} and q satisfy 1
p + 1

q = 1. By direct computations, we have

H1(T ) = µ(θ + 1)−1/qT 1/q := bT 1/q,

H2(T ) = θ(θ − 1/(p− 1))−1/qT−1/p := aT−1/p.

We also denote H(T ) = H1(T ) +H2(T ) and IR(T ) = I1
R(T ) + I2

R(T ).
Let σ > 0 and 0 < ω < 1. We introduce the function

Ψ(σ, ω) ≡ max
x≥0

(σxω − x) = (1− ω)ω
ω

1−ω σ
1

1−ω . (4.2)

Now, we give an upper bound of JR as an integral inequality that plays an
important role in the proof of Theorem 1.1.

Lemma 4.1. Let (u, v) be an L2-solution of (2.1) on [0, Tε). Then we have the
inequality

JR ≤ C1R
sqH(T )q (4.3)

for any T,R > 0 with TR2 < Tε, where s = 2+n
q − 2 and C1 = λ1−q

1 (p− 1)(2/p)q.

Proof. Since (u, v) is an L2-solution on [0, Tε) and ψR ∈ C2
0 ([0, T )×Rn), according

to Proposition 3.2 and TR2 < Tε, by substituting the test function in Definition
3.1 into ψR, we have

λ

∫
[0,TR2)×Rn

(|u|p1 + |v|p2)ψR(t, x) dx dt+ iε

∫
Rn
f(x)ψR(0, x) dx

=

∫
[0,TR2)×Rn

u(−i∂tψR + ∆ψR) dx dt,

(4.4)

and

λ

∫
[0,TR2)×Rn

(|u|p2 + |v|p1)ψR(t, x) dx dt+ iε

∫
Rn
g(x)ψR(0, x) dx

=

∫
[0,TR2)×Rn

v(−i∂tψR + ∆ψR) dx dt.

(4.5)

Taking the real part in (4.4) and (4.5) respectively, we obtain

λ1I
1
R(T )− ε

∫
Rn
f2(x)φR(x) dx = Re

∫
[0,TR2)×Rn

u(−i∂tψR + ∆ψR) dx dt

≤
∫

[0,TR2)×Rn
(|u||∂tψR|+ |u||∆ψR|) dx dt

:= K1
R +K2

R,

(4.6)



10 Y. REN, Y. LI EJDE-2021/24

and

λ1I
2
R(T )− ε

∫
Rn
g2(x)φR(x) dx = Re

∫
[0,TR2)×Rn

v(−i∂tψR + ∆ψR) dx dt

≤
∫

[0,TR2)×Rn
(|v||∂tψR|+ |v||∆ψR|) dx dt

:= K3
R +K4

R.

(4.7)

From these two inequalities we obtain

λ1IR(T ) + JR ≤
4∑
j=1

Kj
R.

Now, estimate the terms Kj
R, j = 1, 2, 3, 4 . A direct calculation yields

∆φR = R−2(∆φ)(x/R),

∂tψR(t, x) = R−2φR(x)(∂tη)(t/R2).

By the above equality, Hölder’s inequality, and noting that p = min{p1, p2}, we
obtain

K1
R =

1

R2

∫
[0,TR2)×Rn

|u|ψ1/p
R η

−1/p
R φ

1/q
R |(∂tη)(t/R2)| dx dt

≤ 1

R2

(∫
[0,TR2)×Rn

|u|pψR dx dt
)1/p

×
(∫

[0,TR2)×Rn
η
−q/p
R φR|(∂tη)(t/R2)|q dx dt

)1/q

≤
(∫

[0,TR2)×Rn
(|u|p1 + |u|p2)ψR dx dt

)1/p

R−2H2(T )R
2+n
q

≤ (I1
R(T ) + I2

R(T ))1/pH2(T )Rs

≤ IR(T )1/pH2(T )Rs.

(4.8)

By (4.1) and Hölder’s inequality, we have

K2
R =

1

R2

∫
[0,TR2)×Rn

|u||∆φR|ηR(t) dx dt

≤ µ 1

R2

∫
[0,TR2)×Rn

|u|ψR dx dt

≤ µ 1

R2

(∫
[0,TR2)×Rn

|u|pψR dx dt
)1/p(∫

[0,TR2)×Rn
ψR dx dt

)1/q

= IR(T )1/pH1(T )Rs.

Similarly, we can obtain

K3
R ≤ IR(T )1/pH2(T )Rs, K4

R ≤ IR(T )1/pH1(T )Rs. (4.9)

Putting (4.8)-(4.9) together, we obtain

λ1IR(T ) + JR ≤ 2RsIR(T )1/pH(T ).

Thus, combining the above inequality with (4.2), noting that λ1 > 0, we have

JR ≤ 2RsH(T )IR(T )1/p − λ1IR(T )



EJDE-2021/24 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 11

≤ λ1Ψ(2H(T )Rs/λ1, 1/p)

= λ1−q
1 (p− 1)(2/p)qRsqH(T )q.

This completes the proof. �

Proof of Theorem 1.1. By changing variables and applying (1.4), we obtain that

JR = −ε
∫
Rn

(f2(x) + g2(x))φR(x) dx

= εRn
∫
Rn
−(f2(Rx) + g2(Rx))φ(x) dx

≥ εRn−kλ−1
1

∫
|x|≥1/R

|x|−kφ(x) dx

≥ εRn−kλ−1
1

∫
|x|≥1/R0

|x|−kφ(x) dx

= CkεR
n−k,

(4.10)

for any R > R0, where 0 < R0 < (b/a)1/2 is a constant and

Ck = λ−1
1

∫
|x|≥1/R0

|x|−kφ(x) dx ≤ λ−1
1

∫
|x|≥1/R0

Rk0φ(x) dx ≤ λ−1
1 Rk0 <∞.

Next, by Corollary 2.4, there exists ε0 > 0 such that Tε > 1 for any ε ∈ (0, ε0). Let
τ ∈ (1, Tε) and R > R0. By using (4.3) with T = τR−2, from (4.10) we deduce
that

ε ≤ C−1
k C1R

sqH(T )qRk−n

= C−1
k C1(aτ−1/pRk/q + bτ1/qR−2+k/q)q.

(4.11)

For each τ ∈ (1, Tε), setting Rτ = (τb/a)1/2 > R0, by substituting R in (4.11) into
Rτ , we have

ε ≤ C−1
k C1

(
aτ−1/p(τb/a)k/2q + bτ1/q(τb/a)−1+k/2q

)q
= C−1

k C12qaq−k/2bk/2τk/2−1/(p−1) = C2τ
−θ,

(4.12)

where θ = 1
p−1 −

k
2 > 0 and C2 = C−1

k C12qaq−k/2bk/2. Since θ > 0, (4.12) yields

τ ≤ Cε−1/θ for arbitrary τ ∈ (1, Tε), with some constant C > 0. Because τ is
arbitrary in (1, Tε), this implies Tε ≤ Cε−1/θ.

Next, we prove (1.5). We suppose that

lim inf
t→T−ε

(‖u(t)‖L2 + ‖v(t)‖L2) < +∞.

Then there exist a sequence {tk}k∈N ⊂ [0, Tε) and a positive constant M > 0 such
that

lim
k→∞

tk = Tε, (4.13)

sup
k∈N

(‖u(tk)‖L2 + ‖v(tk)‖L2) ≤M. (4.14)

On the one hand, by (4.14) and Tε < ∞, there exists a positive constant T (M)
such that we can construct a solution (u, v) of (2.1) that satisfies

u, v ∈ C([tk, tk + T (M));L2)∩Lq1([tk, tk + T (M));Lr1)∩Lq2([tk, tk + T (M));Lr2)



12 Y. REN, Y. LI EJDE-2021/24

for all k ∈ N. On the other hand, by (4.13), when k is sufficiently large, the
inequality tk +T (M) > Tε holds, which contradicts the definition of Tε. Therefore,

lim inf
t→T−ε

(‖u(t)‖L2 + ‖v(t)‖L2) = +∞.

This completes the proof. �

Acknowledgments. This work was supported by the NSFC under grant numbers
11571118, 11771127, and by the Initiated special research for doctoral research
under the grant number GC300501-064.

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EJDE-2021/24 SCHRÖDINGER EQUATIONS WITHOUT GAUGE INVARIANCE 13

Yuanyuan Ren (corresponding author)

School of Computer Science and Technology, Dongguan University of Technology,

Dongguan, Guangdong 523808, China
Email address: renyuanzhpp@163.com

Yongsheng Li
School of Mathematics, South China University of Technology, Guangzhou, Guang-

dong 510640, China

Email address: yshli@scut.edu.cn


	1. Introduction
	2. Local well-posedness
	3. Weak solutions
	4. Proof of main result
	Acknowledgments

	References