The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance

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• 作者：Quanguo Zhang^a ; ^{zhangqg07@163.com" class="auth_mail" title="E-mail the corresponding author} ; Hong-Rui Sun^b ; ^{hrsun@lzu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yaning Li^c ; ^{liyn11@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Time fractional Schrö ; dinger equation ; Nonexistence ; Non-gauge invariance
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：64
• 期：Complete
• 页码：119-124
• 全文大小：750 K

文摘

In this paper, we study the following time fractional Schrödinger equation where 8eee6010953e079aa33" title="Click to view the MathML source">0<α<1$0<\alpha <1$, i^α$i^{\alpha }$ denotes the principal value of i^α$i^{\alpha }$, 8e23d" title="Click to view the MathML source">T>0$T>0$, 999c60c497aa908" title="Click to view the MathML source">λ∈C∖{0}$\lambda \in \mathbb{C}\setminus \left\{0\right\}$, e9d9bde7a444d1e4e8d4d8a" title="Click to view the MathML source">p>1$p>1$, u(t,x)$u(t,x)$ is a complex-valued function, and e8077617b25440d2744a224cc1c946">${}_0^C{D}_t^{\alpha }u$ denotes Caputo fractional derivative of order a9ba690ccbd23a33b4dd0e3d01f40" title="Click to view the MathML source">α.$\alpha .$ We prove that the problem admits no global weak solution with suitable initial data when 8ed5a21ae8e733bba976" title="Click to view the MathML source">1<p<1+2∕N$1<p<1+2∕N$ by using the test function method, and also give some conditions which imply the problem has no global weak solution for every e9d9bde7a444d1e4e8d4d8a" title="Click to view the MathML source">p>1$p>1$.