This paper is concerned with the nonexistence of solutions to the Cauchy problem for the time fractional nonlinear Schrödinger equations with nonlinear memory
where , denotes principal value of , , , , is a complex-valued function, , and are real-valued functions. denotes left Riemann-Liouville fractional integrals of order and
For the nonlinear Schrödinger equations without gauge invariance (i.e. ),
Ikeda and Wakasugi  and Ikeda and Inui   proved blow-up results of solutions for (2) under different conditions for
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  to show the blow-up results.
For nonlinear time fractional Schrödinger equations (i.e., (1) with ), Zhang, Sun and Li  studied the nonexistence of this problem in and proved that the problem admits no global weak solution with suitable initial
data when by using test function method, and also give some
conditions which imply the problem has no global weak solution for every .
In  , Cazenave, Dickstein and Weissler considered a class of heat equation with nonlinear memory. They obtained that the solution blows up in finite time and under suitable conditions the solution exists globally. In  , using test function method, the authors considered a heat equation with nonlinear memory, they determined Fujita critical exponent of the problem.
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
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.
2. Preliminaries and the Main Results
For convenience of statement, let us present some preliminaries that will be used in next sections.
If , and , then we have the following formula of integration by parts
We need calculate Caputo fractional derivative of the following function, which will be used in next sections. For given and , if we let
(see for example  ).
Now, we present the definition of weak solution of (1).
Definition 2.1. Let , and , we call is a weak solution of (1) if
for every with and . Moreover, if can be arbitrarily chosen, then we call u is a global weak solution for of (1).
The following theorems show main result of this paper.
Theorem 2.2. Let . If and satisfies
then problem (1) admits no global weak solution.
Theorem 2.3. If , let . If and satisfies
then problem (1) admits no global weak solution.
3. Proofs of Main Result
In this section, we prove blow-up results and global existence of mild solutions of (1).
Proof of Theorem 2.2. If
for the case , we may as well suppose that and . Let such that for , for and . For , we define
Let . Assuming that u is a weak solution of (1), and since , we have
for some positive constant C independent of T. Then, by (4), (5) and Hölder inequality, we have
Since, we have. Therefore, if the solution of (1) exists globally, then taking, we obtain
which contradicts with the assumption.
For case, we have
Then by a similar proof as above, we can also obtain a contradiction.
Proof of Theorem 2.3. We only consider the case and, since other cases can be proved by a similar method. Take such that
and,. Let. Suppose that u is a bounded weak solution of (1), taking
and define, then using the definition of weak solution of (1) and since, we derive that
by (6) and dominated convergence theorem, let, we have
Hence, by Jensen’s inequality and (7), we have
Denoting, and, then we have
since, we get by taking, which contradicts with the
assumption. Therefore, if is a solution of (1), then.
Supported by NSF of China (11626132, 11601216).
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