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Large-Time Behavior for a Fully Nonlocal Heat Equation

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Abstract

We study the large-time behavior in all Lp norms and in different space-time scales of solutions to a nonlocal heat equation in \(\mathbb {R}^{N}\) involving a Caputo α-time derivative and a power of the Laplacian (−Δ)s, s ∈ (0,1), extending recent results by the authors for the case s = 1. The initial data are assumed to be integrable, and, when required, to be also in Lp. The main novelty with respect to the case s = 1 comes from the behaviour in fast scales, for which, thanks to the fat tails of the fundamental solution of the equation, we are able to give results that are not available neither for the case s = 1 nor, to our knowledge, for the standard heat equation, s = 1, α = 1.

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References

  1. Allen, M., Caffarelli, L., Vasseur, A.: A parabolic problem with a fractional time derivative. Arch. Ration. Mech. Anal. 221, 603–630 (2016)

    Article  MathSciNet  Google Scholar 

  2. Allen, M., Caffarelli, L., Vasseur, A.: Porous medium flow with both a fractional potential pressure and fractional time derivative. Chin. Ann. Math. Ser. B 38, 45–82 (2017)

    Article  MathSciNet  Google Scholar 

  3. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent–II. Geophys. J. Int. R. Astr. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  4. Cartea, Á., del Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general lévy jump distribution functions. Phys. Rev. E 76, 041105 (2007)

    Article  Google Scholar 

  5. Cheng, X., Li, Z., Yamamoto, M.: Asymptotic behavior of solutions to space-time fractional diffusion-reaction equations. Math. Methods Appl. Sci. 40, 1019–1031 (2017)

    Article  MathSciNet  Google Scholar 

  6. Compte, A., Cáceres, M.O.: Fractional dynamics in random velocity fields. Phys. Rev. Lett. 81, 3140–3143 (1998)

    Article  Google Scholar 

  7. Cortazar, C., Quirós, F., Wolanski, N.: A heat equation with memory: large-time behavior. arXiv:2005.02860 (2020)

  8. del Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Fractional diffusion in plasma turbulence. Phys. Plasmas 11, 3854–3864 (2004)

    Article  Google Scholar 

  9. del Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Phys. Rev. Lett. 94, 065003 (2005)

    Article  Google Scholar 

  10. Gorenflo, R., Luchko, Y., Yamamoto, M.: Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18, 799–820 (2015)

    Article  MathSciNet  Google Scholar 

  11. Gripenberg, G.: Volterra integro-differential equations with accretive nonlinearity. J. Differ. Equ. 60, 57–79 (1985)

    Article  MathSciNet  Google Scholar 

  12. Kemppainen, J., Siljander, J., Vergara, V., Zacher, R.: Decay estimates for time-fractional and other non-local in time subdiffusion equations in \(\mathbb {R}^{d}\). Math. Ann. 366, 941–979 (2016)

    Article  MathSciNet  Google Scholar 

  13. Kemppainen, J., Siljander, J., Zacher, R.: Representation of solutions and large-time behavior for fully nonlocal diffusion equations. J. Differ. Equ. 263, 149–201 (2017)

    Article  MathSciNet  Google Scholar 

  14. Kim, K.-H., Lim, S.: Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations. J. Korean Math. Soc. 53, 929–967 (2016)

    Article  MathSciNet  Google Scholar 

  15. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  16. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 777822. C. Cortázar supported by FONDECYT grant 1190102 (Chile). F. Quirós supported by the Spanish Ministry of Science and Innovation, through project MTM2017-87596-P and through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554), and by the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). N. Wolanski supported by CONICET PIP625, Res. 960/12, ANPCyT PICT-2012-0153, UBACYT X117 and MathAmSud 13MATH03 (Argentina).

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Authors and Affiliations

  1. Departamento de Matemática, Pontificia Universidad Católica de Chile, Santiago, Chile

    Carmen Cortázar

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Fernando Quirós

  3. Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UCM-UC3M), 28049, Madrid, Spain

    Fernando Quirós

  4. IMAS-UBA-CONICET, Ciudad Universitaria, Pabellón I, (1428), Buenos Aires, Argentina

    Noemí Wolanski

Authors
  1. Carmen Cortázar
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  2. Fernando Quirós
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  3. Noemí Wolanski
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Correspondence to Fernando Quirós.

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Dedicated to Enrique Zuazua on the occasion of his 60th birthday.

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Cortázar, C., Quirós, F. & Wolanski, N. Large-Time Behavior for a Fully Nonlocal Heat Equation. Vietnam J. Math. 49, 831–844 (2021). https://doi.org/10.1007/s10013-020-00452-w

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  • DOI: https://doi.org/10.1007/s10013-020-00452-w

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