Nonexistence of global solutions for fractional temporal Schrodinger equations and systems
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 lt;1, $i^\alpha$ is the principal value of $i^\alpha$, ${}_0^C D_t^\alpha $ is the Caputo fractional deriv...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2017-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2017/276/abstr.html |
| _version_ | 1818960484547690496 |
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| author | Ibtehal Azman Mohamed Jleli Mokhtar Kirane Bessem Samet |
| author_facet | Ibtehal Azman Mohamed Jleli Mokhtar Kirane Bessem Samet |
| author_sort | Ibtehal Azman |
| collection | DOAJ |
| description | 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 lt;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. |
| first_indexed | 2024-12-20T11:58:16Z |
| format | Article |
| id | doaj.art-fe482ab971d34c43bff3ac114a0b5a8b |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T11:58:16Z |
| publishDate | 2017-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-fe482ab971d34c43bff3ac114a0b5a8b2022-12-21T19:41:36ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912017-11-012017276,117Nonexistence of global solutions for fractional temporal Schrodinger equations and systemsIbtehal Azman0Mohamed Jleli1Mokhtar Kirane2Bessem Samet3 King Saud Univ., Riyadh, Saudi Arabia King Saud Univ., Riyadh, Saudi Arabia Univ. de La Rochelle, Rochelle Cedex, France King Saud Univ., Riyadh, Saudi Arabia 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 lt;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.http://ejde.math.txstate.edu/Volumes/2017/276/abstr.htmlFractional temporal Schrodinger equationnonexistenceglobal weak solution |
| spellingShingle | Ibtehal Azman Mohamed Jleli Mokhtar Kirane Bessem Samet Nonexistence of global solutions for fractional temporal Schrodinger equations and systems Electronic Journal of Differential Equations Fractional temporal Schrodinger equation nonexistence global weak solution |
| title | Nonexistence of global solutions for fractional temporal Schrodinger equations and systems |
| title_full | Nonexistence of global solutions for fractional temporal Schrodinger equations and systems |
| title_fullStr | Nonexistence of global solutions for fractional temporal Schrodinger equations and systems |
| title_full_unstemmed | Nonexistence of global solutions for fractional temporal Schrodinger equations and systems |
| title_short | Nonexistence of global solutions for fractional temporal Schrodinger equations and systems |
| title_sort | nonexistence of global solutions for fractional temporal schrodinger equations and systems |
| topic | Fractional temporal Schrodinger equation nonexistence global weak solution |
| url | http://ejde.math.txstate.edu/Volumes/2017/276/abstr.html |
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