Skip to main content
SpringerLink
Log in

Quantitative uniqueness for fractional heat type operators

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to \((\partial _t - \Delta )^s u =Vu\) for \(s\in [1/2, 1)\) via new Carleman estimates. Our main results Theorems 1.1 and 1.3 can be thought of as a parabolic generalization of the corresponding quantitative uniqueness result in the time independent case due to Rüland and it can also be regarded as a nonlocal generalization of a similar result due to Zhu for solutions to local parabolic equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Arya, V., Banerjee, A., Danielli, D., Garofalo, N.: Space-like strong unique continuation for some fractional parabolic equations. J. Funct. Anal. 284, 109723 (2023)

    MathSciNet  MATH  Google Scholar 

  2. Arya, V., Kumar, D.: Borderline gradient continuity for fractional heat type operators. Proc. R. Soc. Edinburgh Sect. A Math. https://doi.org/10.1017/prm.2022.65

  3. Athanasopoulos, I., Caffarelli, L., Milakis, E.: On the regularity of the non-dynamic parabolic fractional obstacle problem. J. Differ. Equ. 265(6), 2614–2647 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Audrito, A.: On the existence and Hölder regularity of solutions to some nonlinear Cauchy–Neumann problems, arXiv:2107.03308

  5. Audrito, A., Terracini, S.: On the nodal set of solutions to a class of nonlocal parabolic reaction-diffusion equations, arXiv:1807.10135

  6. Bakri, L.: Quantitative uniqueness for Schrödinger operator. Indiana Univ. Math. J. 61(4), 1565–1580 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Balakrishnan, A.: Fractional powers of closed operators and the semigroups generated by them. Pac. J. Math. 10, 419–437 (1960)

    MathSciNet  MATH  Google Scholar 

  8. Banerjee, A., Danielli, D., Garofalo, N., Petrosyan, A.: The regular free boundary in the thin obstacle problem for degenerate parabolic equations. St. Petersburg Math. J. 32(3), 84–126 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Banerjee, A., Danielli, D., Garofalo, N., Petrosyan, A.: The structure of the singular set in the thin obstacle problem for degenerate parabolic equations. Calc. Var. Partial Differ. Equ. 60(3), 91 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Banerjee, A., Davila, G., Sire, Y.: Regularity for parabolic systems with critical growth in the gradient and applications. J. Anal. Math. 146(1), 365–383 (2022)

    MathSciNet  MATH  Google Scholar 

  11. Banerjee, A., Garofalo, N.: Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations. Adv. Math. 336, 149–241 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Banerjee, A., Garofalo, N.: Quantitative uniqueness for elliptic equations at the boundary of \(C^{1, Dini}\) domains. J. Differ. Equ. 261(12), 6718–6757 (2016)

    MATH  Google Scholar 

  13. Banerjee, A., Garofalo, N., Manna, R.: A strong unique continuation property for the heat operator with Hardy type potential. J. Geom. Anal. (2021). https://doi.org/10.1007/s12220-020-00487-y

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Banerjee, N. Garofalo, I. Munive & D. Nhieu, The Harnack inequality for a class of nonlocal parabolic equations, Commun. Contemp. Math. (2021). arXiv:2208.11598

  15. Bellova, K., Lin, F.: Nodal sets of Steklov eigenfunctions. Calc. Var. Partial Differ. Equ. 54(2), 2239–2268 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Biswas, A., Stinga, P.R.: Regularity estimates for nonlocal space-time master equations in bounded domains. J. Evol. Equ. 21, 503–565 (2021)

    MathSciNet  MATH  Google Scholar 

  17. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Dong, H., Phan, T.: Regularity for parabolic equations with singular or degenerate coefficients. Calc. Var. Partial Differ. Equ. 60(1), 44 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93, 161–183 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Donnelly,H., Fefferman, C.: Nodal sets of eigenfunctions: Riemannian manifolds with boundary. Analysis, Et Cetera, Academic Press, Boston, pp. 251–262 (1990)

  21. Escauriaza, L., Fernandez, F.: Unique continuation for parabolic operators (English summary). Ark. Mat. 41(1), 35–60 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Escauriaza, L., Fernandez, F., Vessella, S.: Doubling properties of caloric functions. Appl. Anal. 85(1–3), 205–223 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Escauriaza, L., Vessella, S.: Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, in: Inverse Problems: Theory and Applications, Cortona/Pisa, 2002, in: Contemp. Math. Amer. Math. Soc., Providence, RI, vol. 333, pp. 79–87 (2003)

  24. Fall, M., Felli, V.: Unique continuation property and local asymptotics of solutions to fractional elliptic equations. Commun. Partial Differ. Equ. 39, 354–397 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Garofalo, N.: Two classical properties of the Bessel quotient \(I_{\nu +1}/I_\nu \) and their implications in pde’s. Advances in harmonic analysis and partial differential equations, Contemp. Math., vol. 748, pp. 57–97, Amer. Math. Soc., Providence, RI (2020)

  26. Garofalo, N., Lin, F.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35, 245–268 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Garofalo, N., Lin, F.: Unique continuation for elliptic operators: a geometric-variational approach. Commun. Pure Appl. Math. 40, 347–366 (1987)

    MathSciNet  MATH  Google Scholar 

  28. Han, Q., Lin, F.: Nodal sets of solutions of parabolic equations .II. Commun. Pure Appl. Math. 47, 1219–1238 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Hyder, A., Segatti, A., Sire, Y., Wang, C.: Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary. Commun. Partial Differ. Equ. 47(9), 1845–1882 (2022)

    MathSciNet  MATH  Google Scholar 

  30. Jones, B.F.: Lipschitz spaces and the heat equation. J. Math. Mech. 18, 379–409 (1968)

    MathSciNet  MATH  Google Scholar 

  31. Jones, B.F.: A fundamental solution for the heat equation which is supported in a strip. J. Math. Anal. Appl. 60, 314–324 (1977)

    MathSciNet  MATH  Google Scholar 

  32. Kenig, C.E.: Some recent applications of unique continuation. Recent Dev. Nonlinear Partial Differ. Equ. Contemp. Math. 439, 25 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Kukavica, I.: Quantitative uniqueness for second order elliptic operators. Duke Math. J. 91, 225–240 (1998)

    MathSciNet  MATH  Google Scholar 

  34. Kukavica, I.: Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg–Landau equation. Electron. J. Differ. Equ. 61, 15 (2000)

    MathSciNet  MATH  Google Scholar 

  35. Lai, R., Lin, Y., Ruland, A.: The Calderón problem for a space-time fractional parabolic equation. SIAM J. Math. Anal. 52(3), 2655–2688 (2020)

    MathSciNet  MATH  Google Scholar 

  36. Lieberman, G.: Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, xii+439 pp. ISBN: 981-02-2883-X (1996)

  37. Lin, F.: Nodal sets of solutions of elliptic equations of elliptic and parabolic equations. Commun. Pure Appl. Math. 44, 287–308

  38. Logunov, A.: Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture. Ann. Math. 187, 241–262 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Logunov, A.: Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure. Ann. Math. 187, 221–239 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Logunov, A., Malinnikova, E.: Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, 50 years with Hardy spaces, Oper. Theory Adv. Appl , pp. 333–344 (2018)

  41. Litsgärd, M., Nyström, K.: On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients. J. Evol. Equ. 23(1), 3 (2023)

    MathSciNet  MATH  Google Scholar 

  42. Meshov, V.: On the possible rate of decrease at infinity of the solutions of second-order partial differential equations. Math. USSR-Sb. 72(2), 343–361 (1992)

    MathSciNet  Google Scholar 

  43. Nyström, K., Sande, O.: Extension properties and boundary estimates for a fractional heat operator. Nonlinear Anal. 140, 29–37 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Poon, C.C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21(3–4), 521–539 (1996)

    MathSciNet  MATH  Google Scholar 

  45. Rüland, A.: On quantitative unique continuation properties of fractional Schrodinger equations: doubling, vanishing order and nodal domain estimates. Trans. Am. Math. Soc. 369, 2311–2362 (2017)

    MathSciNet  MATH  Google Scholar 

  46. Rüland, A.: Unique continuation for fractional Schrödinger equations with rough potentials. Commun. Partial Differ. Equ. 40(1), 77–114 (2015)

    MATH  Google Scholar 

  47. Rüland, A., Salo, M.: The fractional Calderón problem: low regularity and stability. Nonlinear Anal. 193, 111529 (2020)

    MathSciNet  MATH  Google Scholar 

  48. Sampson, C.H.: A characterization of parabolic Lebesgue spaces. Thesis (Ph.D.)—Rice University, pp. 91 (1968)

  49. Samko, S.G.: Hypersingular integrals and their applications. Analytical Methods and Special Functions, 5. Taylor & Francis Group, London, pp. xviii+359 (2002)

  50. Stinga, P.R., Torrea, J.L.: Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation. SIAM J. Math. Anal. 49, 3893–3924 (2017)

    MathSciNet  MATH  Google Scholar 

  51. Vessella, S.: Carleman estimates, optimal three cylinder inequality, and unique continuation properties for solutions to parabolic equations. Commun. Partial Differ. Equ. 28, 637–676 (2003)

    MathSciNet  MATH  Google Scholar 

  52. Yau, S.T.: Seminar on differential geometry. Princeton University Press, Princeton, vol. 102 (1982)

  53. Zhu, J.: Quantitative uniqueness for elliptic equations. Am. J. Math. 138, 733–762 (2016)

    MathSciNet  MATH  Google Scholar 

  54. Zhu, J.: Quantitative uniqueness of solutions to parabolic equations. J. Funct. Anal. 275(9), 2373–2403 (2018)

    MathSciNet  MATH  Google Scholar 

  55. Zhu, J.: Doubling property and vanishing order of Steklov eigenfunctions. Commun. Partial Differ. Equ. 40(8), 1498–1520 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

A.B is supported in part by SERB Matrix grant MTR/2018/000267 and by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520.

Author information

Authors and Affiliations

  1. Centre For Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore, 560065, India

    Vedansh Arya & Agnid Banerjee

Authors
  1. Vedansh Arya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Agnid Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Agnid Banerjee.

Additional information

Communicated by Susanna Terracini.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arya, V., Banerjee, A. Quantitative uniqueness for fractional heat type operators. Calc. Var. 62, 195 (2023). https://doi.org/10.1007/s00526-023-02535-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-023-02535-1

Mathematics Subject Classification

Search

Navigation