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Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity


Author: Takahisa Inui
Journal: Proc. Amer. Math. Soc. 144 (2016), 2901-2909
MSC (2010): Primary 35Q55
DOI: https://doi.org/10.1090/proc/12938
Published electronically: March 18, 2016
MathSciNet review: 3487223
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Abstract: We consider the following semirelativistic nonlinear Schrödinger equation (SNLS):

$\displaystyle \left \{ \begin {array}{ll} i\partial _t u \pm (m^2-\Delta )^{1/2... ...imes \mathbb{R}^d, \\ u(0,x)=u_0(x), & x \in \mathbb{R}^d, \end{array} \right .$    

where $ m\geq 0$, $ \lambda \in \mathbb{C} \setminus \{ 0\}$, $ d\in \mathbb{N}$, $ T>0$, and $ \partial _t=\partial /\partial t$. Here $ (m^2-\Delta )^{1/2}:=\mathcal {F}^{-1} (m^2+\vert\xi \vert^2 )^{1/2} \mathcal {F}$, where $ \mathcal {F}$ denotes the Fourier transform. Fujiwara and Ozawa proved the nonexistence of global weak solutions to SNLS for some initial data in the case of $ d=1$, $ m=0$, and $ 1<p\leq 2$ by a test function method. In this paper, we extend their result to a more general setting: for example, $ m\geq 0$, $ d\in \mathbb{N}$, or $ p>1$. Moreover, we obtain the upper estimates of weak solutions to SNLS. The key to the proof is to choose an appropriate test function.

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Additional Information

Takahisa Inui
Affiliation: Department of Mathematics, Kyoto University, Kyoto 60-5802, Japan
Email: inui@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/proc/12938
Keywords: Semirelativistic equations, test function, nonexistence of global solution
Received by editor(s): March 19, 2015
Published electronically: March 18, 2016
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society