Asymptotic behaviour for the fractional heat equation in the Euclidean space

Abstract

We consider weak solutions of the fractional heat equation posed in the whole n-dimensional space, and establish their asymptotic convergence to the fundamental solution as under the assumption that the initial datum is an integrable function, or a finite Radon measure. Convergence with suitable rates is obtained for solutions with a finite first initial moment, while for solutions with compactly supported initial data convergence in relative error holds. The results are applied to the fractional Fokker–Planck equation. Brief mention of other techniques and related equations is made.

Keywords:

• Fractional heat equation
• convergence to the fundamental solution

AMS Subject Classifications:

• 35B40
• 35R11
• 35K55

Acknowledgements

The author would like to thank the hospitality of the Mathematical Institute of the University of Warwick. I thank R. Zacher for information about his paper [35] and the anonymous referee for an interesting suggestion.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Ministerio de Economía y Competitividad [Project MTM2014-52240-P] (Spain).