Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

Abstract

We consider the problem

$$\begin{aligned} (P)\left\{ \begin{array}{llll} u_t+(-\Delta )^{s} u &{}=&{} \lambda \dfrac{u^p}{\delta ^{2s}(x)} &{} \quad \text { in }\Omega _{T}\equiv \Omega \times (0,T) , \\ u(x,0)&{}=&{}u_0(x) &{} \quad \text { in }\Omega , \\ u&{}=&{}0 &{}\quad \text { in } ({I\!\!R}^N\setminus \Omega ) \times (0,T), \end{array}\right. \end{aligned}$$

where \(\Omega \subset {I\!\!R}^N\) is a bounded regular domain (in the sense that \(\partial \Omega \) is of class \({\mathcal {C}}^{0,1}\)), \(\delta (x)=\text {dist}(x,\partial \Omega )\), \(0<s<1\), \(p>0\), \(\lambda >0\). The purpose of this work is twofold. First We analyze the interplay between the parameters sp and \(\lambda \) in order to prove the existence or the nonexistence of solution to problem (P) in a suitable sense. This extends previous similar results obtained in the local case \(s=1\). Second We will especially point out the differences between the local and nonlocal cases.

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Notes

  1. 1.

    This fractional Laplacian operator is sometimes called the restricted fractional Laplacian (see for example [15, 16, 49]), or regional fractional Laplacian (see for instance [43, 48]) or Dirichlet fractional Laplacian (see [44]).

  2. 2.

    The choice of the constant \(a_{N,s}\) is motivated, among others, by the following assertion:

    $$\begin{aligned} \lim _{s\rightarrow 0^+}(-\Delta )^s u= u\quad \text {and}\quad \lim _{s\rightarrow 1^-}(-\Delta )^s u= -\Delta u \end{aligned}$$

    where \(\Delta \) is the classical Laplacian. For a proof, see for instance [30, Proposition 4.4]).

  3. 3.

    This question was first posed by H. Brezis and J.-L. Lions in the early 1980s.

  4. 4.

    For the ease of the reader, let us recall that we denote by \(\Lambda _s(\Omega )\) the best possible constant in Hardy’s inequality (see Theorem 1.1, Remark 1.2 ).

References

  1. 1.

    N. Abatangelo: Large Solutions for Fractional Laplacian Operators, PhD thesis, 2015. https://arxiv.org/pdf/1511.00571.pdf.

  2. 2.

    N. Abatangelo: Very large solutions for the fractional Laplacian: towards a fractional Keller-Osserman condition. Adv. Nonlinear Anal. 6 (2017), no. 4, 383-405.

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    N. Abatangelo: Large s-Harmonic functions with boundary blow-up solutions for the fractional Laplacian. Discrete and Continuous Dynamical systems 35, No 12 (2015), 5555-5607.

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    B. Abdellaoui, K. Biroud, J. Davila and F. Mahmoudi: Nonlinear elliptic problem related to the Hardy inequality with singular term at the boundary. Commun. Contemp. Math. 17(2015), no. 3, 1450033, 28 pp.

  5. 5.

    B. Abdellaoui, K. Biroud, A. Primo: A semilinear parabolic problem with singular term at the boundary. Journal of Evolution Equations, Volume (16), 2016, 131–153.

  6. 6.

    B. Abdellaoui, K. Biroud, A. Primo: Nonlinear fractional elliptic problem with singular term at the boundary. Complex Var. Elliptic Equ. 64 (2019), no. 6, 909–932.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    B. Abdellaoui, M. Medina, I. Peral, A. Primo: Optimal results for the fractional heat equation involving the Hardy potential. Nonlinear Analysis TMA. 140 (2016), 166-207

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    B. Abdellaoui, I. Peral, A. Primo and F. Soria : On the KPZ equation with fractional diffusion. arXiv:1904.04593.pdf

  9. 9.

    R. A. Adams: Sobolev spaces. Academic Press, New York, 1975.

    Google Scholar 

  10. 10.

    Adimurth, J. Giacomoni, S. Santra, Positive solutions to a fractional equation with singular nonlinearity, J. Diff. Equaitions 265, (2018), no 4, 1191-1226

  11. 11.

    A. Attar, S. Merchan, I. Peral: A remark on existence of semilinear heat equation involving a Hardy-Leray potential. Journal of Evolution Equations, Volume 15 (1),(2015) 239–250.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    P. Baras and J. Goldstein: The heat equation with a singular potential. Trans. Amer. Math. Soc. 294 (1984), 121–139.

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    B. Barrios, M. Medina: Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, Proceedings of the Royal Society of Edinburgh, page 1 of 21 https://doi.org/10.1017/prm.2018.77

  14. 14.

    K. Bogdan, B. Dyda: The best constant in a fractional Hardy inequality. Math. Nachrichten, 284 (2011), no (5-6), 629–638.

  15. 15.

    M. Bonforte, A. Figalli, J. L. Vázquez: Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. PDE 11 (2018), no. 4, 945–982.

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    M. Bonforte, Y. Sire, J. L. Vázquez: Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains. Disrete Contin. Dyn. Syst. -A 35(2015), no. 12, 5725–5767.

  17. 17.

    M. Bonforte, Y. Sire, J. L. Vázquez: Optimal existence and uniqueness theory for the fractional heat equation. Nonline Analysis T.M.A. 153, (2017), 142-168

  18. 18.

    M. Bonforte, J. L. Vázquez: A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains. Arch. Rat. Mec. Anal. 218, no. 1 (2015), 317–362.

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    L. Brasco, E. Cinti: On the fractional Hardy inequalities in convex sets. DCDS. No. 38, no. 8 (2018), 4019- 4040.

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    H. Brezis, X. Cabré: Some simple nonlinear PDE’s without solution. Boll. Unione. Mat. Ital. Sez. B, 8(1998), 223–262.

    MathSciNet  MATH  Google Scholar 

  21. 21.

    H. Brezis, M. Marcus: Hardy’s inequalities revisited, Ann. Sc. Norm. Sup. Pisa 25 (1997), 217-237.

    MathSciNet  MATH  Google Scholar 

  22. 22.

    H. Brezis, M. Marcus, I. Shafrir: Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000), 177–191.

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    C. Bucur, E. Valdinoci: Nonlocal Diffusion and Applications. Lectures Notes of Unione Matematica Italiana. Vol. 20 (2016), Springer.

  24. 24.

    X. Cabré, Y. Martel: Existence versus explosion instantanTe pour des Tquations de la chaleur linTaires avec potentiel singulier. C. R. Acad. Sci. Paris, STr. I Math. 329 (1999), no. 11, 973–978.

  25. 25.

    L. Caffarelli, L. Silvestre: An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32 (2007), no. 7-9, 1245–1260.

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    L. Caffarelli, Y. Sire: On Some Pointwise Inequalities Involving Nonlocal Operators, Harmonic Analysis, Partial Differential Equations and Applications. Applied and Numerical Harmonic Analysis. BirkhSuser (2017), 1–18.

  27. 27.

    A. Capella, J. Dávila, L. Dupaigne, Y. Sire: Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Partial Differential Equations, 365 (8) (2011), 1353–1384.

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    H. Chen H, L. Veron: Semilinear fractional elliptic equations involving measures. J. Diff. Equations, no. 257 (2014), 1457–1486.

  29. 29.

    A. Cordoba, D. Cordoba: A pointwise estimate for fractionary derivatives with appl ications to partial differential equations. Proceedings of the National Academy of Sciences of the Unite d States of America 100 (26) (2003), 15316-15317

    MATH  Article  Google Scholar 

  30. 30.

    E. Di Nezza, G. Palatucci, E. Valdinoci: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136 (2012), no. 5, 521–573.

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    S. Dipierro, O. Savin, E. Valdinoci: All functions are locally s-harmonic up to a small error. J. Eur. Math. Soc. (JEMS) 19, no. 4 (2017), 957–966.

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    B. Dyda: A fractional order Hardy inequality. J. Math. Ill., 48 (2) (2004), 575–588,

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    X. Fernandez-Real, X. Ros-Oton: Boundary regularity for the fractional heat equation. Revista de la Real Academia de Ciencias Exactas, Ffsicas y Naturales. Serie A. Matemáticas. March 2016, V. 110, No.1, 49–64.

  34. 34.

    S. Filippas, L. Moschini, A. Tertikas: Sharp Trace Hardy-Sobolev-Mazýa Inequalities and the Fractional Laplacian. Arch. Rational Mech. Anal. 208 (2013), 109–161.

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    J. Garcia-Azorero, I. Peral: Hardy inequalities and some critical elliptic and parabolic problems , J. Differential Equations, 144 (1998), 441–476.

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    J. Giacomoni, T. Mukherjee, K. Sreenadh: Positive solutions of fractional elliptic equation with critical and singular nonlinearity. Adv. Nonlinear Anal. 6 (2017), no. 3, 327–354.

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    A. Kufner, L. E. Persson: Weighted inequalities of Hardy type. World Scientific Publishing 2003.

  38. 38.

    T. Leonori, I. Peral, A. Primo, F. Soria: Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 6031–6068.

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    J.-L. Lions, E. Magenes: ProblFmes aux Limites Non HomgFnes et Applications. Vol. 1. no. 17. Paris, Dunod (1968).

  40. 40.

    Y. Martel: Complete blow up and global behavior of solution of\(u_t-\Delta u=g(u)\). Ann. Inst. Henri PoincarT, (15) 6 (1998), 687–723.

  41. 41.

    T. Matskewich, P. E. Sobolevskii: The best possible constant in generalized Hardy’s inequality for convex domain in\(I\!\!R^N\). Nonlinear Anal, Theory, Methods and Appl. (29)(1997), 1601–1610.

  42. 42.

    D. S. Mitrinovic, J. E. Pecaric, A. M. Fink: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and its Applications, 1991.

  43. 43.

    G. Molica Bisci, V. D. Radulescu, R. Servadei: Variational Methods For Nonlocal Fractional Problems, Encyclopedia of Mathematics and Its Applications 162, Cambridge University Press.

  44. 44.

    R. Musina, A. Nazarov: On fractional Laplacians. Comm. Partial Differential Equations 39 (2014), no. 9, 1780–1790.

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    A. C. Ponce: Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016.

    Google Scholar 

  46. 46.

    R. Servadei, E. Valdinoci: On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh, Sec. A, 144 (2014), no. 4, 831–855. 389 (2012), 887-898.

  47. 47.

    X. Ros-Oton, J. Serra: The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math.Pures Appl. 101 (2014), 275-302

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    R. Servadei, E. Valdinoci: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appli. 389 (2012), 887–898.

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    J. L. Vázquez: The mathematical theories of diffusion. Nonlinear and fractional diffusion, in “Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions”, Springer Lecture Notes in Mathematics, ISBN 978-3-319-61493-9. C.I.M.E. Foundation Subseries; course held in Cetraro, Italy 2016.

  50. 50.

    J. L. Vázquez, E. Zuazua: The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential. J. Funct. Anal. 173 (2000), no. 1, 103–153.

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewer for his/her careful reading of our manuscript and his/her many insightful comments and suggestions. Part of this work was realized while the first and the second authors were visiting the “Institut Elie Cartan”, Université de Lorraine. They would like to thank the Institute for the warm hospitality. B. Abdellaoui is partially supported by project MTM2016-80474-P, MINECO, Spain and by the DGRSDT, Algeria.

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Correspondence to El-Haj Laamri.

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Abdellaoui, B., Biroud, K. & Laamri, EH. Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00623-9

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Mathematics Subject Classification

  • 35B05
  • 35K15
  • 35B40
  • 35K55
  • 35K65

Keywords

  • Fractional Nonlinear parabolic problems
  • Singular Hardy potential
  • Complete blow-up results