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On finite time blow-up for the mass-critical Hartree equations

Published online by Cambridge University Press:  03 June 2015

Yonggeun Cho ,
Gyeongha Hwang ,
Soonsik Kwon  and
Sanghyuk Lee
Yonggeun Cho
Affiliation:
Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, Republic of Korea, ([email protected])
Gyeongha Hwang*
Affiliation:
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 689-798, Republic of Korea, ([email protected])
Soonsik Kwon
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea, ([email protected])
Sanghyuk Lee
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea, ([email protected])
*
*Corresponding author
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Abstract

We consider the fractional Schrödinger equations with focusing Hartree-type nonlinearities. When the energy is negative, we show that the solution blows up in a finite time. For this purpose, based on Glassey’s argument, we obtain a virial-type inequality.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 467 - 479
Copyright
Copyright © Royal Society of Edinburgh 2015 

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References

1 Cazenave, T.. Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol. 10 (Providence, RI: American Mathematical Society, 2003).Google Scholar
2 Cho, Y. and Nakanishi, K.. On the global existence of semirelativistic Hartree equations. RIMS Kôkyûroku Bessatsu B22 (2010), 145–166.Google Scholar
3 Cho, Y., Ozawa, T., Sasaki, H. and Shim, Y.. Remarks on the semirelativistic Hartree equations. Discrete Contin. Dynam. Syst. A23 (2009), 1273–1290.Google Scholar
4 Christ, F. M. and Weinstein, M. I.. Dispersion of small amplitude solution of the generalized Korteweg–de Vries equation. J. Funct. Analysis 100 (1991), 87–109.Google Scholar
5 Fröhlich, J. and Lenzmann, E.. Blow-up for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60 (2007), 1691–1705.Google Scholar
6 Glassey, R. T.. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18 (1977), 1794–1797.Google Scholar
7 Stein, E. M.. Singular integrals and differentiability properties of functions (Princeton University Press, 1970).Google Scholar
8 Stein, E. M.. Harmonic analysis (Princeton University Press, 1993).Google Scholar
9 Stein, E. M. and Weiss, G.. Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7 (1958), 503–514.Google Scholar