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The heat equation for the Dirichlet fractional Laplacian with negative potentials: Existence and blow-up of nonnegative solutions

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Abstract

We establish conditions ensuring either existence or blow-up of nonnegative solutions for the heat equation generated by the Dirichlet fractional Laplacian perturbed by negative potentials on bounded sets. The elaborated theory is supplied by some examples.

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References

  1. Beldi, A., Belhajrhouma, N., Ben Amor, A.: Pointwise estimates for the ground states of singular Dirichlet fractional Laplacian. J. Phys. A: Math. Theor., 46(44), 445201 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bogdan, K., Byczkowski, T., Kulczycki, T., et al.: Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009

    Book  Google Scholar 

  3. Baras, P., Goldstein, J. A.: The heat equation with a singular potential. Trans. Amer. Math. Soc., 284(1), 121–139 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogdan, K., Grzywny, T., Ryznar, M.: Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab., 38(5), 1901–1923 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cabré, X., Martel, Y.: Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier. C. R. Acad. Sci. Paris Sér. I Math., 329(11), 973–978 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Sylvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Diff. Eq., 32(8), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Z. Q., Song, R.: Hardy inequality for censored stable processes. Tohoku Math. J. (2), 55(3), 439–450 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davies, E. B., Simon, B.: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal., 59(2), 335–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dubkov, A. A., Spagnolo, B., Uchaikin, V. V.: Lévy flight superdiffusion: an introduction. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18(9), 2649–2672 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Delfour, M. C., Zolésio, J. P.: Shape analysis via oriented distance functions. J. Funct. Anal., 123(1), 129–201 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Edmunds, D. E, Evans, W. D.: Hardy Operators, Function Spaces and Embeddings, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004

    Book  Google Scholar 

  12. Frank, R. L., Lieb, E. H., Seiringer, R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc., 21(4), 925–950 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fukushima, M., Oshima, Y., Masayoshi, T.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics, extended Walter de Gruyter & Co. Berlin, 2011

    MATH  Google Scholar 

  14. Goldstein, G. R., Goldstein, J. A., Rhandi, A.: Weighted Hardy’s inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl. Anal., 91(11), 2057–2071 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldstein, A. J., Zhang, Q. S.: Linear parabolic equations with strong singular potentials. Trans. Amer. Math. Soc., 355(1), 197–211 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Humphries, N. E., Weimerskirch, H., Queiroz, N., et al.: Foraging success of biological Lévy flights recorded in situ. PNAS, 109(19), 7169–7174 (2012)

    Article  Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition

    Book  Google Scholar 

  18. Keller, M., Lenz, D., Vogt, H., et al.: Note on basic features of large time behaviour of heat kernels. J. Reine und Angew. Math., 708, 73–95 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Kulczycki, T.: Intrinsic ultracontractivity for symmetric stable processes. Bull. Polish Acad. Sci. Math., 46(3), 325–334 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Ishige, K., Ishiwata, M.: Heat equation with a singular potential on the boundary and the Kato inequality. J. Anal. Math., 118(1), 161–176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stollmann, P.: Admissible and regular potentials for Schrödinger forms. J. Operator Theory, 18(1), 139–151 (1987)

    MathSciNet  MATH  Google Scholar 

  22. Voigt, J.: Absorption semigroups, their generators, and Schrödinger semigroups. J. Funct. Anal., 67(2), 167–205 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Gabès, University Gabès, Gabès, 6072, Tunisia

    Ali Ben Amor

  2. Department of Mathematics, Faculty of Sciences of Tunis, University Tunis ElManar, Tunis, 1001, Tunisia

    Tarek Kenzizi

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  1. Ali Ben Amor
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  2. Tarek Kenzizi
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Correspondence to Ali Ben Amor.

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Ben Amor, A., Kenzizi, T. The heat equation for the Dirichlet fractional Laplacian with negative potentials: Existence and blow-up of nonnegative solutions. Acta. Math. Sin.-English Ser. 33, 981–995 (2017). https://doi.org/10.1007/s10114-017-6246-8

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  • DOI: https://doi.org/10.1007/s10114-017-6246-8

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