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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si2.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=681a282cc7d7f8eee6010953e079aa33" title="Click to view the MathML source">0<α<1class="mathContainer hidden">class="mathCode">$0<\alpha <1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si3.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=adc2bf51547ea834cb3bc4e3a04873c3" title="Click to view the MathML source">i^αclass="mathContainer hidden">class="mathCode">$i^{\alpha }$ denotes the principal value of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si3.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=adc2bf51547ea834cb3bc4e3a04873c3" title="Click to view the MathML source">i^αclass="mathContainer hidden">class="mathCode">$i^{\alpha }$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si5.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=7f15483e27ddba27149a756271b8e23d" title="Click to view the MathML source">T>0class="mathContainer hidden">class="mathCode">$T>0$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si6.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=038244209bf7cfa77999c60c497aa908" title="Click to view the MathML source">λ∈C∖{0}class="mathContainer hidden">class="mathCode">$\lambda \in \mathbb{C}\setminus \left\{0\right\}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si7.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=956550b56e9d9bde7a444d1e4e8d4d8a" title="Click to view the MathML source">p>1class="mathContainer hidden">class="mathCode">$p>1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si8.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=7ea6478c4050413c30c1ff81f9cbfdbc" title="Click to view the MathML source">u(t,x)class="mathContainer hidden">class="mathCode">$u(t,x)$ is a complex-valued function, and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si9.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=4be8077617b25440d2744a224cc1c946">class="imgLazyJSB inlineImage" height="16" width="35" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S089396591630252X-si9.gif">class="mathContainer hidden">class="mathCode">${}_0^C{D}_t^{\alpha }u$ denotes Caputo fractional derivative of order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si10.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=11fa9ba690ccbd23a33b4dd0e3d01f40" title="Click to view the MathML source">α.class="mathContainer hidden">class="mathCode">$\alpha .$ We prove that the problem admits no global weak solution with suitable initial data when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si11.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=de773f122f478ed5a21ae8e733bba976" title="Click to view the MathML source">1<p<1+2∕Nclass="mathContainer hidden">class="mathCode">$1<p<1+2∕N$ by using the test function method, and also give some conditions which imply the problem has no global weak solution for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089396591630252X&_mathId=si7.gif&_user=111111111&_pii=S089396591630252X&_rdoc=1&_issn=08939659&md5=956550b56e9d9bde7a444d1e4e8d4d8a" title="Click to view the MathML source">p>1class="mathContainer hidden">class="mathCode">$p>1$.