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The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance
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In this paper, we study the following time fractional Schrödinger equation where mathmlsrc">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<1mathContainer hidden">mathCode"><math id="mml2" overflow="scroll" altimg="si2.gif">0<α<1math>, mathmlsrc">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">iαmathContainer hidden">mathCode"><math id="mml3" overflow="scroll" altimg="si3.gif">iαmath> denotes the principal value of mathmlsrc">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">iαmathContainer hidden">mathCode"><math id="mml4" overflow="scroll" altimg="si3.gif">iαmath>, mathmlsrc">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>0mathContainer hidden">mathCode"><math id="mml5" overflow="scroll" altimg="si5.gif">T>0math>, mathmlsrc">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}mathContainer hidden">mathCode"><math id="mml6" overflow="scroll" altimg="si6.gif">λmathvariant="double-struck">C{0}math>, mathmlsrc">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>1mathContainer hidden">mathCode"><math id="mml7" overflow="scroll" altimg="si7.gif">p>1math>, mathmlsrc">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)mathContainer hidden">mathCode"><math id="mml8" overflow="scroll" altimg="si8.gif">u(t,x)math> is a complex-valued function, and mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math id="mml9" overflow="scroll" altimg="si9.gif">0CDtαumath> denotes Caputo fractional derivative of order mathmlsrc">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.mathContainer hidden">mathCode"><math id="mml10" overflow="scroll" altimg="si10.gif">α.math> We prove that the problem admits no global weak solution with suitable initial data when mathmlsrc">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∕NmathContainer hidden">mathCode"><math id="mml11" overflow="scroll" altimg="si11.gif">1<p<1+2Nmath> by using the test function method, and also give some conditions which imply the problem has no global weak solution for every mathmlsrc">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>1mathContainer hidden">mathCode"><math id="mml12" overflow="scroll" altimg="si7.gif">p>1math>.

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