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The time fractional Schrödinger equation with a nonlinearity of Hartree type

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Abstract

The aim of this work is to show existence, uniqueness and regularity properties of local in time mild solutions for the nonlinear space-time fractional Schrödinger equation (1.1), with a fractional time derivative of order \(\alpha \in (0,1),\) and with a nonlinear term of Hartree type.

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References

  1. Bergh, J., Löfström, J., Interpolation Spaces: An Introduction, New York: Springer (1976).

    Book  Google Scholar 

  2. Cho, Y., Ozawa, T. Global solutions of semirelativistic Hartree type equations. J. Korean Math. Soc.  44, no. 5, 1065-1078 (2007).

    Article  MathSciNet  Google Scholar 

  3. Cho, Y., Hajaiej, H., Hwang, G., Ozawa, T., On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity. Funkcial. Ekvac.  56, 193–224 (2013).

    Article  MathSciNet  Google Scholar 

  4. Cho, Y., Hwang, G., Kwon, S., Lee, S., On finite time blow-up for the mass-critical Hartree equations. Proc. Roy. Soc. Edinburgh Sect. A  145, 467–479 (2015).

    Article  MathSciNet  Google Scholar 

  5. Frohlich, J., Lenzmann, E. Mean field Limit of Quantum Bose Gases and Nonlinear Hartree Equation. Séminaire Équations aux dérivées partielles (Polytechnique), Exposé. arXiv:math-ph/0409019 (2004).

  6. Frohlich, J., Lars, B., Jonsson, G., Lenzmann, E. Boson Stars as Solitary Waves. Commun. Math. Phys.  274, 1–30 (2007).

    Article  MathSciNet  Google Scholar 

  7. Górka, P., Prado, H., Trujillo, J. The time fractional Schrödinger equation on Hilbert space. Integral Equations Operator Theory.  87, 1–14 (2017).

    Article  MathSciNet  Google Scholar 

  8. Grafakos, L. Classical Fourier Analysis, Third edition, Graduate Texts in Math.  249, New York: Springer (2014).

    Google Scholar 

  9. Grafakos, L., Oh, S. The Kato-Ponce inequality. Comm. Partial Differential Equations.  39, 1128-1157 (2014).

    Article  MathSciNet  Google Scholar 

  10. Herr, S., Tesfahun, A. Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations.  259, 5510–5532 (2015).

    Article  MathSciNet  Google Scholar 

  11. Iomin, A. Fractional-time quantum dynamics, Physical Review E 80, 022103 (2009).

    Article  Google Scholar 

  12. Iomin, A. Fractional evolution in quantum mechanics, Chaos, Solitons & Fractals: X, 1, 100001, (2019).

    Google Scholar 

  13. Iomin, A. Fractional time quantum mechanics. Handbook of Fractional Calculus with Applications, volume 4: Application in Physics, part A; Tarasov, V. (Ed). Series edited by J.A. Tenreiro-Machad . De Gruyter, Berlin/Boston (2019).

    Google Scholar 

  14. Kwasnicki, M. Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal.  20, 7–51 (2017).

    Article  MathSciNet  Google Scholar 

  15. Laskin, N. Fractional quantum mechanics and Lévy path integrals. Physics Letters A. 268, 298–305 (2002).

    Article  MathSciNet  Google Scholar 

  16. Laskin, N. Time fractional quantum mechanics. Chaos Solitons Fractals, 1–13 (2017).

    MATH  Google Scholar 

  17. Lenzmann, E. Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 10 (1), 43–64 (2007).

    Article  MathSciNet  Google Scholar 

  18. Lenzmann, E., Lewin, M. On singularity formation for the \(L^2\)-critical Boson star equation. Nonlinearity.  24, 3515–3540 (2011).

    Article  MathSciNet  Google Scholar 

  19. Podlubny, I. Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.  198, Slovak Republic: Academic Press (1998).

    MATH  Google Scholar 

  20. Reed, M., Simon, B. Methods of Modern Mathematical Physics I: Functional Analysis. Revised and enlarged edition, Academic Press, an imprint of Elsevier (1980).

    Google Scholar 

  21. Stein, E. M., Murphy, T. S. Harmonic Analysis: Real variable methods, orthogonality and oscillatory integrals  3, New Jersey: Princeton University press (1993).

    Google Scholar 

  22. Tao, T. Nonlinear Dispersive Equations: local and Global Analysis  106, Rhode Island: American Mathematical Soc. (2006).

    Book  Google Scholar 

  23. Taylor, M. Partial Differential Equations III. , New York: Springer (2010).

    Google Scholar 

  24. Treves, F. Topological Vector Spaces, Distributions and Kernels: Pure and Applied Mathematics  25, New York: Elsevier (2016).

    Google Scholar 

  25. Wu, D. Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity. J. Math. Anal. Appl. Elsevier  411, 530–542 (2014).

    Article  MathSciNet  Google Scholar 

  26. Zhang, J., Zhu, S. Stability of standing waves for the nonlinear fractional Schrödinger equation. Dynam. Differential Equations 29 (3), 1017–1030 (2017).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

H.P. has been partially supported by FONDECYT grant #1170571.

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  1. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 165, Est. Central, Santiago, Chile

    Humberto Prado & José Ramírez

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  1. Humberto Prado
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  2. José Ramírez
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Correspondence to Humberto Prado.

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Prado, H., Ramírez, J. The time fractional Schrödinger equation with a nonlinearity of Hartree type. J. Evol. Equ. 21, 1845–1864 (2021). https://doi.org/10.1007/s00028-020-00658-y

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  • DOI: https://doi.org/10.1007/s00028-020-00658-y

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