University of Oulu

Kemppainen, J., Zacher, R. (2019) Long-time behavior of non-local in time Fokker–Planck equations via the entropy method. Mathematical Models and Methods in Applied Sciences, 29 (02), 209-235. https://doi.org/10.1142/S0218202519500076

Long-time behavior of non-local in time Fokker–Planck equations via the entropy method

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Author: Kemppainen, Jukka1; Zacher, Rico2
Organizations: 1Applied and Computational Mathematics, University of Oulu, Pentti Kaiteran Katu 1, P. O. Box 8000, FI-90014 Oulu, Finland
2Institute of Applied Analysis, Ulm University, 89069 Ulm, Germany
Format: article
Version: accepted version
Access: embargoed
Persistent link: http://urn.fi/urn:nbn:fi-fe201903057154
Language: English
Published: World Scientific, 2019
Publish Date: 2020-01-31
Description:

Abstract

We consider a rather general class of non-local in time Fokker–Planck equations and show by means of the entropy method that as t→∞, the solution converges in L1 to the unique steady state. Important special cases are the time-fractional and ultraslow diffusion case. We also prove estimates for the rate of decay. In contrast to the classical (local) case, where the usual time derivative appears in the Fokker–Planck equation, the obtained decay rate depends on the entropy, which is related to the integrability of the initial datum. It seems that higher integrability of the initial datum leads to better decay rates and that the optimal decay rate is reached, as we show, when the initial datum belongs to a certain weighted L2 space. We also show how our estimates can be adapted to the discrete-time case thereby improving known decay rates from the literature.

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Series: Mathematical models & methods in applied sciences
ISSN: 0218-2025
ISSN-E: 0218-2025
ISSN-L: 0218-2025
Volume: 29
Issue: 2
Pages: 209 - 235
DOI: 10.1142/S0218202519500076
OADOI: https://oadoi.org/10.1142/S0218202519500076
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Subjects:
Copyright information: Electronic version of an article published as Mathematical Models and Methods in Applied Sciences, VOL. 29, NO. 02 2019, 209-235 doi:10.1142/S0218202519500076, © World Scientific Publishing Company, https://doi.org/10.1142/S0218202519500076