Electron. J. Differential Equations, Vol. 2017 (2017), No. 276, pp. 1-17.

Nonexistence of global solutions for fractional temporal Schrodinger equations and systems

Ibtehal Azman, Mohamed Jleli, Mokhtar Kirane, Bessem Samet

Abstract:
We, first, consider the nonlinear Schrodinger equation
$$ i^\alpha {}_0^C D_t^\alpha u+\Delta u= \lambda |u|^p+\mu a(x)\cdot\nabla |u|^q, \quad t>0,\; x\in \mathbb{R}^N, $$
where $0<\alpha<1$, $i^\alpha$ is the principal value of $i^\alpha$, ${}_0^C D_t^\alpha $ is the Caputo fractional derivative of order $\alpha$, $\lambda\in \mathbb{C}\backslash\{0\}$, $\mu\in \mathbb{C}$, $p>q>1$, $u(t,x)$ is a complex-valued function, and $a: \mathbb{R}^N\to \mathbb{R}^N$ is a given vector function. We provide sufficient conditions for the nonexistence of global weak solution under suitable initial data. Next, we extend our study to the system of nonlinear coupled equations
$$\displaylines{ i^\alpha {}_0^C D_t^\alpha u+\Delta u = \lambda |v|^p+\mu a(x)\cdot\nabla |v|^q, \quad t>0,\;x\in \mathbb{R}^N,\cr i^\beta {}_0^C D_t^\beta v+\Delta v = \lambda |u|^\kappa+\mu b(x)\cdot\nabla |u|^\sigma, \quad t>0,\; x\in \mathbb{R}^N, }$$
where $0<\beta\leq \alpha<1$, $\lambda\in \mathbb{C}\backslash\{0\}$, $\mu\in \mathbb{C}$, $p>q>1$, $\kappa>\sigma>1$, and $a,b: \mathbb{R}^N\to \mathbb{R}^N$ are two given vector functions. Our approach is based on the test function method.

Submitted October 14, 2017. Published November 8, 2017.
Math Subject Classifications: 4735, 26A33.
Key Words: Fractional temporal Schrodinger equation; nonexistence; global weak solution.

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  Ibtehal Azman
Department of Mathematics
King Saud University, P.O. Box 2455
Riyadh, 11451, Saudi Arabia
email: ibtehalazman@yahoo.com
Mohamed Jleli
Department of Mathematics
King Saud University, P.O. Box 2455,
Riyadh, 11451, Saudi Arabia
email: jleli@ksu.edu.sa
Mokhtar Kirane
LaSIE, Faculté des Sciences
Pole Sciences et Technologies, Université de La Rochelle
Avenue M. Crepeau, 17042 La Rochelle Cedex, France
email: mkirane@univ-lr.fr
Bessem Samet
Department of Mathematics
King Saud University, P.O. Box 2455,
Riyadh, 11451, Saudi Arabia
email: bsamet@ksu.edu.sa

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