The Nonexistence of Global Solutions for a Time Fractional Schrödinger Equation with Nonlinear Memory

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**Keywords:**

1. Introduction

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

$\{\begin{array}{l}{i}^{\alpha}{}_{0}{}^{C}D{}_{t}^{\alpha}u+\Delta u=\lambda {}_{0}I{}_{t}^{1-\gamma}\left({\left|u\right|}^{p}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {\mathbb{R}}^{N},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\hfill \\ u\left(0,x\right)=g\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {\mathbb{R}}^{N},\hfill \end{array}$ (1)

where $0<\alpha <\gamma <1$ , ${i}^{\alpha}$ denotes principal value of ${i}^{\alpha}$ , $p>1$ , $T>0$ , $\lambda ={\lambda}_{1}+{\lambda}_{2}i\in \u2102\backslash \left\{0\right\},{\lambda}_{1},{\lambda}_{2}\in \mathbb{R}$ , $u=u\left(t,x\right)$ is a complex-valued function, $g\left(x\right)={g}_{1}\left(x\right)+{g}_{2}\left(x\right)i$ , ${g}_{1}\left(x\right)$ and ${g}_{2}\left(x\right)$ are real-valued functions. ${}_{0}I{}_{t}^{1-\gamma}$ denotes left Riemann-Liouville fractional integrals of order $1-\gamma $ and

${}_{0}{}^{C}D{}_{t}^{\alpha}u=\frac{\partial}{\partial t}{}_{0}I{}_{t}^{1-\alpha}\left(u\left(t,x\right)-u\left(0,x\right)\right)$ .

For the nonlinear Schrödinger equations without gauge invariance (i.e. $\alpha =\gamma =1$ ),

$\{\begin{array}{l}i{u}_{t}+\Delta u=\lambda {\left|u\right|}^{p},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {\mathbb{R}}^{N},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\hfill \\ u\left(0,x\right)=g\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {\mathbb{R}}^{N},\hfill \end{array}$ (2)

Ikeda and Wakasugi [1] and Ikeda and Inui [2] [3] proved blow-up results of solutions for (2) under different conditions for

$1<p<1+\frac{2}{N}$ and $1<p<1+\frac{4}{N}$ .

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 [4] to show the blow-up results.

For nonlinear time fractional Schrödinger equations (i.e., (1) with $\gamma =1$ ), Zhang, Sun and Li [12] studied the nonexistence of this problem in ${C}_{0}\left({R}^{N}\right)$ and proved that the problem admits no global weak solution with suitable initial

data when $1<p<1+\frac{2}{N}$ by using test function method, and also give some

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

In [13] , 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 [14] , 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

$1<p<1+\frac{2\left(\alpha +1-\gamma \right)}{\alpha N}$ and $1<p<1+\frac{1-\gamma}{\alpha}$ .

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 ${}_{0}{}^{C}D{}_{t}^{\alpha}f\in {L}^{1}\left(\mathrm{0,}T\right)$ , $g\in {C}^{1}\left(\left[\mathrm{0,}T\right]\right)$ and $g\left(T\right)=0$ , then we have the following formula of integration by parts

${\int}_{0}^{T}}g{}_{0}{}^{C}D{}_{t}^{\alpha}f\text{d}t={\displaystyle {\int}_{0}^{T}}\left(f\left(t\right)-f\left(0\right)\right){}_{t}{}^{C}D{}_{T}^{\alpha}g\text{d}t.$ (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

$\phi \left(t\right)=\{\begin{array}{l}{\left(1-\frac{t}{T}\right)}^{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}t\le T,\hfill \\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>T,\hfill \end{array}$

then

${}_{t}{}^{C}D{}_{T}^{\alpha}\phi \left(t\right)=\frac{\Gamma \left(n+1\right)}{\Gamma \left(n+1-\alpha \right)}{T}^{-\alpha}{\left(1-\frac{t}{T}\right)}^{n-\alpha},\text{\hspace{0.17em}}t\le T,$

(see for example [15] ).

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

Definition 2.1. Let $g\in {L}_{loc}^{1}\left({R}^{N}\right)$ , $0<\alpha <\gamma <1$ and $T>0$ , we call $u\in {L}^{p}\left(\left(\mathrm{0,}T\right)\mathrm{,}{L}_{loc}^{\infty}\left({R}^{N}\right)\right)$ is a weak solution of (1) if

${\int}_{{R}^{N}}}{\displaystyle {\int}_{0}^{T}}\lambda {}_{0}I{}_{t}^{1-\gamma}\left({\left|u\right|}^{p}\right)\phi +{i}^{\alpha}g\left(x\right){}_{t}{}^{C}D{}_{T}^{\alpha}\phi \text{d}t\text{d}x={\displaystyle {\int}_{{R}^{N}}}{\displaystyle {\int}_{0}^{T}}u\left(\Delta \phi +{i}^{\alpha}{}_{t}{}^{C}D{}_{T}^{\alpha}\phi \right)\text{d}t\text{d}x$

for every $\phi \in {C}_{x\mathrm{,}t}^{\mathrm{2,1}}\left({R}^{N}\times \left[\mathrm{0,}T\right]\right)$ with $sup{p}_{x}\phi \subset \subset {R}^{N}$ and $\phi \left(x,T\right)=0$ . Moreover, if $T>0$ can be arbitrarily chosen, then we call u is a global weak solution for of (1).

Denote

${G}_{1}\left(x\right)=\mathrm{cos}\frac{\text{\pi}\alpha}{2}{g}_{1}\left(x\right)-\mathrm{sin}\frac{\text{\pi}\alpha}{2}{g}_{2}\left(x\right)$ , ${G}_{2}\left(x\right)=\mathrm{cos}\frac{\text{\pi}\alpha}{2}{g}_{2}\left(x\right)+\mathrm{sin}\frac{\text{\pi}\alpha}{2}{g}_{1}(x)$

and $\beta =1-\gamma $ .

The following theorems show main result of this paper.

Theorem 2.2. Let $1<p<1+\frac{2\left(\alpha +\beta \right)}{\alpha N}$ . If $g\in {L}^{1}\left({\mathbb{R}}^{N}\right)$ and satisfies

${\lambda}_{1}{\displaystyle {\int}_{{\mathbb{R}}^{N}}}{G}_{1}\left(x\right)\text{d}x>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda}_{2}{\displaystyle {\int}_{{\mathbb{R}}^{N}}}{G}_{2}\left(x\right)\text{d}x>0,$

then problem (1) admits no global weak solution.

Theorem 2.3. If $1<p<1+\frac{\beta}{\alpha}$ , let $\chi \left(x\right)={\left({\displaystyle {\int}_{{\mathbb{R}}^{N}}}{\text{e}}^{-\sqrt{{N}^{2}+{\left|x\right|}^{2}}}\text{d}x\right)}^{-1}{\text{e}}^{-\sqrt{{N}^{2}+{\left|x\right|}^{2}}}$ . If $g\in {L}_{\left({\mathbb{R}}^{N}\right)}^{1}$ and satisfies

${\lambda}_{1}{\displaystyle {\int}_{{\mathbb{R}}^{N}}}{G}_{1}\left(x\right)\chi \left(x\right)\text{d}x>\mathrm{0,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda}_{2}{\displaystyle {\int}_{{\mathbb{R}}^{N}}}{G}_{2}\left(x\right)\chi \left(x\right)\text{d}x>\mathrm{0,}$

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

$1<p<1+\frac{2\left(\alpha +\beta \right)}{\alpha N}$ ,

for the case ${\lambda}_{1}{\displaystyle {\int}_{{\mathbb{R}}^{N}}}{G}_{1}\left(x\right)\text{d}x>0$ , we may as well suppose that ${\lambda}_{1}>0$ and ${\int}_{{\mathbb{R}}^{N}}}{G}_{1}\left(x\right)\text{d}x>0$ . Let $\Phi \in {C}_{0}^{\infty}\left({\mathbb{R}}^{N}\right)$ such that $\Phi \left(s\right)=1$ for $\left|s\right|\le 1$ , $\Phi \left(s\right)=0$ for $\left|s\right|>2$ and $0\le \Phi \left(s\right)\le 1$ . For $T>0$ , we define

${\phi}_{1}\left(x\right)={\left(\Phi \left({T}^{-\frac{\alpha}{2}}\left|x\right|\right)\right)}^{\frac{2p}{p-1}},\text{\hspace{0.17em}}{\phi}_{2}\left(t\right)={\left(1-\frac{t}{T}\right)}^{m},\text{\hspace{0.17em}}m\ge \mathrm{max}\left\{1,\frac{p\left(\alpha +\beta \right)}{p-1}\right\},\text{\hspace{0.17em}}t\in \left[0,T\right].$

Let $\phi \left(x,t\right)={}_{t}{}^{C}D{}_{T}^{\beta}{\phi}_{1}\left(x\right){\phi}_{2}\left(t\right)$ . Assuming that u is a weak solution of (1), and since $\alpha +\beta <1$ , we have

that is

(4)

Note that

(5)

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

Hence

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

(6)

Since

and

by (6) and dominated convergence theorem, let, we have

(7)

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

Denoting, and, then we have

Thus,

So,

,

since, we get by taking, which contradicts with the

assumption. Therefore, if is a solution of (1), then.

Supported

Supported by NSF of China (11626132, 11601216).

References

[1] Ikeda, M. and Wakasugi, Y. (2013) Small Data Blow-Up of L2-Solution for the Nonlinear Schrödinger Equation without Gauge Invariance. Differential Integral Equations, 26, 1275-1285.

[2] Ikeda, M. and Inui, T. (2015) Small Data Blow-Up of L2 or H1-Solution for the Semilinear Schrodinger Equation without Gauge Invariance. Journal of Evolution Equations, 15, 1-11.

https://doi.org/10.1007/s00028-015-0273-7

[3] Ikeda, M. and Inui, T. (2015) Some Non-Existence Results for the Semilinear Schrödinger Equation without Gauge Invariance. Journal of Mathematical Analysis and Applications, 425, 758-773.

https://doi.org/10.1016/j.jmaa.2015.01.003

[4] Mitidieri, E. and Pohozaev, S.I. (2001) A Priori Estimates and Blow-Up of Solutions to Nonlinear Partial Differential Equations and Inequalities. Proceedings of the Steklov Institute of Mathematics, 234, 1-383.

[5] Mainardi, F. (1994) On the Initial Value Problem for the Fractional Diffusion-Wave Equation. In Rionero, S. and Ruggeri, T., Eds., Waves and Stability in Continuous Media, World Scientific, Singapore, 246-251.

[6] Nigmatullin, R.R. (1986) The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry. Physica Status Solidi, 133, 425-430.

https://doi.org/10.1002/pssb.2221330150

[7] Andrade, B. and Viana, A. (2017) On a Fractional Reaction-Diffusion Equation. Zeitschrift für angewandte Mathematik und Physik, 68, 59.

https://doi.org/10.1007/s00033-017-0801-0

[8] Li, Y.N. (2015) Regularity of Mild Solutions for Fractional Abstract Cauchy Problem with Order . Zeitschrift für angewandte Mathematik und Physik, 66, 3283-3298.

https://doi.org/10.1007/s00033-015-0577-z

[9] Li, Y.N., Sun, H.R. and Feng, Z.S. (2016) Fractional Abstract Cauchy Problem with Order . Dynamics of PDE, 13, 155-177.

[10] Vergara, V. and Zacher, R. (2017) Stability, Instability, and Blowup for Time Fractional and Other Nonlocal in Time Semilinear Subdiffusion Equations. Journal of Evolution Equations, 17, 599-626.

https://doi.org/10.1007/s00028-016-0370-2

[11] Zhang, Q.G. and Sun, H.R. (2015) The Blow-Up and Global Existence of Solutions of Cauchy Problems for a Time Fractional Diffusion Equation. Topological Methods in Nonlinear Analysis, 46, 69-92.

https://doi.org/10.12775/TMNA.2015.038

[12] Zhang, Q.G., Sun, H.R. and Li, Y.N. (2017) The Nonexistence of Global Solutions for a Time Fractional Nonlinear Schrödinger Equation without Gauge Invariance. Applied Mathematics Letters, No. 64, 119-124.

https://doi.org/10.1016/j.aml.2016.08.017

[13] Cazenave, T., Dickstein, F. and Weissler, F.B. (2008) An Equation Whose Fujita Critical Exponent Is Not Given by Scaling. Nonlinear Analysis, 68, 862-874.

https://doi.org/10.1016/j.na.2006.11.042

[14] Fino, A.Z. and Kirane, M. (2012) Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation. Quarterly of Applied Mathematics, 70, 133-157.

https://doi.org/10.1090/S0033-569X-2011-01246-9

[15] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, Vol 204. Elsevier Science B.V., Amsterdam.

https://doi.org/10.1016/S0304-0208(06)80001-0