^{1}

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In this paper, we study the nonexistence of solutions of the following time fractional nonlinear Schr
?dinger equations with nonlinear memory
where 0<λ<β<1
, ι^{λ}
denotes the principal value of ι^{λ}
, p>1
, T>0
, λ∈C/{0}
, u(t,x)
is a complex-value function,
denotes left Riemann-Liouville fractional integrals of order 1-λ
and
is the Caputo fractional derivative of order
. We obtain that the problem admits no global weak solution when
and
under different conditions for initial data.

This paper is concerned with the nonexistence of solutions to the Cauchy problem for the time fractional nonlinear Schrödinger equations with nonlinear memory

{ i α D 0 C t α u + Δ u = λ I 0 t 1 − γ ( | u | p ) , x ∈ ℝ N , t > 0 , u ( 0 , x ) = g ( x ) , x ∈ ℝ N , (1)

where 0 < α < γ < 1 , i α denotes principal value of i α , p > 1 , T > 0 , λ = λ 1 + λ 2 i ∈ ℂ \ { 0 } , λ 1 , λ 2 ∈ ℝ , u = u ( t , x ) is a complex-valued function, g ( x ) = g 1 ( x ) + g 2 ( x ) i , g 1 ( x ) and g 2 ( x ) are real-valued functions. I 0 t 1 − γ denotes left Riemann-Liouville fractional integrals of order 1 − γ and

D 0 C t α u = ∂ ∂ t I 0 t 1 − α ( u ( t , x ) − u ( 0 , x ) ) .

For the nonlinear Schrödinger equations without gauge invariance (i.e. α = γ = 1 ),

{ i u t + Δ u = λ | u | p , x ∈ ℝ N , t > 0 , u ( 0 , x ) = g ( x ) , x ∈ ℝ N , (2)

Ikeda and Wakasugi [

1 < p < 1 + 2 N and 1 < p < 1 + 4 N .

The main tool they used is test function method. This method is based on rescalings of a compactly support test function to prove blow-up results which is first used by Mitidieri and Pohozaev [

Recently, it has been seen that fractional differential equations have better effects in many realistic applications than the classical ones. So, considerable attention has been attracted to time fractional diffusion equation which arises in electromagnetic, acoustic and mechanical phenomena etc. [

For nonlinear time fractional Schrödinger equations (i.e., (1) with γ = 1 ), Zhang, Sun and Li [

data when 1 < p < 1 + 2 N by using test function method, and also give some

conditions which imply the problem has no global weak solution for every p > 1 .

In [

Motivated by above results, in present paper, our purpose is to study the nonexistence of global weak solutions of (1) with a condition related to the sign of initial data when

1 < p < 1 + 2 ( α + 1 − γ ) α N and 1 < p < 1 + 1 − γ α .

This paper is organized as follows. In Section 2, some preliminaries and the main results are presented. In Section 3, we give proof of the main results.

For convenience of statement, let us present some preliminaries that will be used in next sections.

If D 0 C t α f ∈ L 1 ( 0, T ) , g ∈ C 1 ( [ 0, T ] ) and g ( T ) = 0 , then we have the following formula of integration by parts

∫ 0 T g D 0 C t α f d t = ∫ 0 T ( f ( t ) − f ( 0 ) ) D t C T α g d t . (3)

We need calculate Caputo fractional derivative of the following function, which will be used in next sections. For given T > 0 and n > 0 , if we let

φ ( t ) = { ( 1 − t T ) n , t ≤ T , 0 , t > T ,

then

D t C T α φ ( t ) = Γ ( n + 1 ) Γ ( n + 1 − α ) T − α ( 1 − t T ) n − α , t ≤ T ,

(see for example [

Now, we present the definition of weak solution of (1).

Definition 2.1. Let g ∈ L l o c 1 ( R N ) , 0 < α < γ < 1 and T > 0 , we call u ∈ L p ( ( 0, T ) , L l o c ∞ ( R N ) ) is a weak solution of (1) if

∫ R N ∫ 0 T λ I 0 t 1 − γ ( | u | p ) φ + i α g ( x ) D t C T α φ d t d x = ∫ R N ∫ 0 T u ( Δ φ + i α D t C T α φ ) d t d x

for every φ ∈ C x , t 2,1 ( R N × [ 0, T ] ) with s u p p x φ ⊂ ⊂ R N and φ ( x , T ) = 0 . Moreover, if T > 0 can be arbitrarily chosen, then we call u is a global weak solution for of (1).

Denote

G 1 ( x ) = cos π α 2 g 1 ( x ) − sin π α 2 g 2 ( x ) , G 2 ( x ) = cos π α 2 g 2 ( x ) + sin π α 2 g 1 (x)

and β = 1 − γ .

The following theorems show main result of this paper.

Theorem 2.2. Let 1 < p < 1 + 2 ( α + β ) α N . If g ∈ L 1 ( ℝ N ) and satisfies

λ 1 ∫ ℝ N G 1 ( x ) d x > 0 , or λ 2 ∫ ℝ N G 2 ( x ) d x > 0 ,

then problem (1) admits no global weak solution.

Theorem 2.3. If 1 < p < 1 + β α , let χ ( x ) = ( ∫ ℝ N e − N 2 + | x | 2 d x ) − 1 e − N 2 + | x | 2 . If g ∈ L ( ℝ N ) 1 and satisfies

λ 1 ∫ ℝ N G 1 ( x ) χ ( x ) d x > 0, or λ 2 ∫ ℝ N G 2 ( x ) χ ( x ) d x > 0,

then problem (1) admits no global weak solution.

In this section, we prove blow-up results and global existence of mild solutions of (1).

Proof of Theorem 2.2. If

1 < p < 1 + 2 ( α + β ) α N ,

for the case λ 1 ∫ ℝ N G 1 ( x ) d x > 0 , we may as well suppose that λ 1 > 0 and ∫ ℝ N G 1 ( x ) d x > 0 . Let Φ ∈ C 0 ∞ ( ℝ N ) such that Φ ( s ) = 1 for | s | ≤ 1 , Φ ( s ) = 0 for | s | > 2 and 0 ≤ Φ ( s ) ≤ 1 . For T > 0 , we define

φ 1 ( x ) = ( Φ ( T − α 2 | x | ) ) 2 p p − 1 , φ 2 ( t ) = ( 1 − t T ) m , m ≥ max { 1 , p ( α + β ) p − 1 } , t ∈ [ 0 , T ] .

Let φ ( x , t ) = D t C T β φ 1 ( x ) φ 2 ( t ) . Assuming that u is a weak solution of (1), and since α + β < 1 , we have

that is

Note that

for some positive constant C independent of T. Then, by (4), (5) and Hölder inequality, we have

Hence

Since

which contradicts with the assumption.

For case

Then by a similar proof as above, we can also obtain a contradiction.

Proof of Theorem 2.3. We only consider the case

and

and define

Since

and

by (6) and dominated convergence theorem, let

Hence, by Jensen’s inequality and (7), we have

Denoting

Thus,

So,

since

assumption. Therefore, if

Supported by NSF of China (11626132, 11601216).

Li, Y.N. and Zhang, Q.G. (2018) The Nonexistence of Global Solutions for a Time Fractional Schrödinger Equation with Nonlinear Memory. Journal of Applied Mathematics and Physics, 6, 1418-1424. https://doi.org/10.4236/jamp.2018.67118